^AHvaan-] 


:«ARK^.     ^E-UBRARYOc 


.vsnig  %\*n 

&& 

™& 


lOS-ANCflft 


|gp| 


^^>< 
^toaWWHW?1 


^WWW^ 

SOr^l 


%B3NV-SOl^ 


a*1 


, 


F-CALI 


mo- 

•CALIF 


MECHANICS  AND  HEAT 


-V 


^Me  Qraw-MlBock  (n.  7m 

PUBLISHERS     OF     BOOKS      F  O  R^ 

Coal  Age  v  Electric  Railway  Journal 
Electrical  World  v  Engineering  News-Record 
American  Machinist  v  Ingenieria  Internacional 
Engineering g Mining  Journal  ^  Power 
Chemical  6  Metallurgical  Engineering 
Electrical  Merchandising 


PHYSICS     FOR    TECHNICAL     STUDENTS 

MECHANICS 

AND 

HEAT 


BY 
WILLIAM  BALLANTYNE  ANDERSON,  PH.  D. 

ASSOCIATE    PROFESSOR   OP   PHT8IC8,    IOWA   STATE   COLLEGE 


FIRST  EDITION 
SIXTH  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC. 
NEW  YORK:  370  SEVENTH  AVENUE 

LONDON:  6  &  8  BOUVERIE  ST.,  E.  C.  4 
1914 


COPYRIGHT,  1914,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


PREFACE 

/n 

The  present  volume  is  the  outgrowth  of  mimeograph  notes 
which  the  author  has  used  in  connection  with  a  course  of  lectures 
given  during  the  past  six  years.  Since  the  author  has  also  con- 
ducted the  recitations  for  several  sections  during  this  time,  the 

1  successive  revisions  of  the  notes  have  been  made  by  one  viewing 
the  work  from  two  angles,  that  of  class  instructor,  as  well  as  that 
of  lecturer.  It  is  believed  that  in  this  way  a  keener  realization 

>^  of  the  student's  difficulties,  and  a  better  appreciation  of  what 
parts  should  be  revised,  have  been  obtained  than  would  have 

fc.     been  possible  without  this  two-fold  contact. 

^{  We  now  have  a  large  and  rapidly  increasing  number  of  students 
who  are  interested  primarily  in  the  practical  side  of  education. 

^  With  the  needs  of  these  students  in  mind,  the  practical  side  of  the 
subject  has  been  emphasized  throughout  the  book.  This 
method,  it  is  believed,  will  sustain  interest  in  the  subject  by 
showing  its  application  to  everyday  affairs,  and  will,  it  is  hoped, 
be  appreciated  by  .both  students  and  instructors  in  Agriculture 
and  Engineering.  In  this  connection,  attention  is  directed  to 
sections  18,  19,  20,  29,  30,  39,  44,  54,  56,  60,  62,  63,  76,  80,  83, 
108,  109,  111,  134,  138,  170,  185,  189,  190,  195,  200,  204,  205, 
206,  218  and  Chapters  VII,  XII,  XVII,  and  XVIII. 

More  space  than  usual  has  been  devoted  to  the  treatment  of 
Force,  Torque,  Translatory  Motion,  and  Rotary  Motion.  It  is 
felt  that  the  great  importance  of  these  topics,  which  underlie 
so  much  of  the  subsequent  work  of  the  student,  warrants  such 
treatment.  Probably  everyone  who  has  taught  the  theory  of 
electrical  measuring  instruments,  for  example,  has  realized  that 
the  student's  greatest  handicap  is  the  lack  of  a  thorough  grasp  of 
the  fundamental  principles  of  mechanics.  The  student  who  has 
'^thoroughly  mastered  elementary  mechanics  has  done  much 
toward  preparing  himself  for  effective  work  in  technical  lines. 

The  sketches,  which  are  more  numerous  than  is  usual  in  such  a 
text,  are  chosen  with  special  reference  to  the  help  they  will  be  in 
enabling  the  student  to  readily  grasp  important  or  difficult 
principles.  Wherever  possible,  every  principle  involved  in  the 

v 

443914 


vi  PREFACE 

text  is  brought  up  again  in  a  problem;  so  that  in  working  all  of 
the  problems  a  review  of  practically  the  entire  book  is  obtained. 
For  a  complete  course,  the  text  should  be  accompanied  by  lectures 
and  laboratory  work. 

In  the  treatment  of  many  of  the  subjects,  the  author  is  indebted 
to  various  authors  of  works  in  Physics,  among  whom  may  be 
mentioned  Professors  Spinney,  Duff,  Watson,  and  Crew.  The 
order  in  which  the  different  subjects  are  treated  is  that  which 
seems  most  logical  and  most  teachable,  and  was  given  much 
thought. 

Thanks  are  due  Professor  G.  M.  Wilcox,  of  the  Department  of 
Physics,  Armour  Institute,  and  Professor  W.  Weniger,  of  the 
Department  of  Physics  at  Oregon  Agricultural  College,  for  their 
careful  reading  of  the  original  mimeograph  notes  and  for  the 
numerous  suggestions  which  they  offered.  I  wish  also  to  thank 
my  colleagues,  Professor  H.  J.  Plagge  and  Professor  W.  Kunerth, 
for  reading  of  the  manuscript  and  proofs,  and  for  valuable  sug- 
gestions. Thanks  are  also  due  to  Professor  W.  R.  Raymond  of 
the  English  Department  of  this  College  for  reading  much  of 
the  manuscript  during  revision,  and  to  Professor  J.  C.  Bowman 
of  the  same  department,  for  reading  practically  all  of  the  manu- 
script just  before  it  went  to  press. 

IOWA  STATE  COLLEGE,  W   B   A 

March,  1914.  W.J5.  A. 


CONTENTS 

PAGE 
PREFACE    v 

PART  I 

MECHANICS 

CHAPTER  I 

MEASUREMENT 1 

Section  1.  The  three  fundamental  quantities.  2.  Units  and 
numerics.  3.  Fundamental  units.  4.  Standards  of  length,  mass, 
and  time.  5.  The  metric  system.  6.  Conversion  of  units.  7. 
Measurement  of  length.  8.  The  vernier  caliper.  9.  The  mi- 
crometer caliper.  10.  The  micrometer  microscope.  11.  Meas- 
urement of  mass,  inertia.  12.  Measurement  of  time. 

CHAPTER  II 

VECTORS 11 

Section  13.  Scalars  and  vectors  denned.  14.  Representation  of 
vectors  by  straight  lines.  15.  Addition  of  vectors,  resultant. 
16.  The  vector  polygon.  17.  Vectors  in  equilibrium.  18.  The 
crane.  19.  Resolution  of  vectors  into  components.  20.  Sailing 
against  the  wind.  21.  Sailing  faster  than  the  wind. 

CHAPTER  III 

TRANSLATORY  MOTION 23 

Section  22.  Kinds  of  motion.  23.  Speed,  average  speed,  velocity 
and  average  velocity.  24.  Acceleration.  25.  Accelerating  force. 
26.  Uniform  motion  and  uniformly  accelerated  motion.  27. 
Universal  gravitation.  28.  The  law  of  the  inverse  square  of  the 
distance.  29.  Planetary  motion.  30.  The  tides.  31.  Accelera- 
tion of  gravity  and  accelerating  force  in  free  fall.  32.  Units  of 
weight  and  units  of  force,  compared.  33.  Motion  of  falling 
bodies;  velocity — initial,  final  and  average.  34.  Distance  fallen 
in  a  given  time.  35.  Atwood's  machine.  36.  Motion  of  projec- 
tiles; initial  velocity  vertical.  37.  Motion  of  projectiles;  initial 
velocity  horizontal.  38.  Motion  of  projectiles;  initial  velocity 
inclined.  39.  Time  of  flight  and  range  of  a  projectile.  40. 
Spring  gun  experiment.  41.  The  plotting  of  curves.  42.  New- 
ton's three  laws  of  motion.  43.  Action  and  reaction,  inertia 


viii  CONTENTS 

PAGE 

force,  principle  of  d'Alembert.  44.  Practical  applications  of 
reaction.  45.  Momentum,  impulse,  impact  and  conservation  of 
momentum.  46.  The  ballistic  pendulum. 

CHAPTER  IV 

ROTARY  MOTION 59 

Section  47.  Kinds  of  rotary  motion.  48.  Torque.  49.  Resultant 
torque  and  an tiresultant  torque.  50.  Angular  measurement.  51. 
Angular  velocity  and  angular  acceleration.  52.  Relation  between 
linear  and  angular  velocity  and  acceleration.  53.  The  two  condi- 
tions of  equilibrium  of  a  rigid  body.  54.  Moment  of  inertia  and 
accelerating  torque.  55.  Value  and  unit  of  moment  of  inertia.  56. 
Use  of  the  flywheel.  57.  Formulas  for  translatory  and  rotary 
motion  compared. 

CHAPTER  V 

UNIFORM  CIRCULAR  MOTION,  SIMPLE  HARMONIC  MOTION 72 

Section  58.  Central  and  centrifugal  forces  and  radial  acceleration. 
59.  Bursting  of  emery  wheels  and  flywheels.  60.  The  cream 
separator.  61.  Efficiency  of  cream  separator.  62.  Elevation  of 
the  outer  rail  on  curves  in  a  railroad  track.  63.  The  centrifugal 
governor.  63a.  The  gyroscope.  64.  Simple  harmonic  motion. 
65.  Acceleration  and  force  of  restitution  in  S.H.M.  66.  Period 
in  S.H.M.  67.  The  simple  gravity  pendulum.  68.  The  torsion 
pendulum. 

CHAPTER  VI 

WORK,  ENERGY,  AND  POWER 89 

Section  69.  Work.  70.  Units  of  work.  71.  Work  done  if  the  line 
of  motion  is  not  in  the  direction  of  the  applied  force.  72.  Work 
done  by  a  torque.  73.  Energy — potential  and  kinetic.  74.  Trans- 
formation and  conservation  of  energy.  75.  Value  of  potential  and 
kinetic  energy  in  work  units.  76.  Energy  of  a  rotating  body.  77. 
Dissipation  of  energy.  78.  Sliding  friction.  79.  Coefficient  of 
friction.  80.  Rolling  friction.  81.  Power.  82.  Units  of  power. 
83.  Prony  brake. 

CHAPTER  VII 

MACHINES 110 

Section  84.  Machine  defined.  85.  Mechanical  advantage  and 
efficiency.  86.  The  simple  machines.  87.  The  lever.  88.  The 
pulley.  89.  The  wheel  and  axle.  90.  The  inclined  plane.  91. 
The  wedge.  92.  The  screw.  93.  The  chain  hoist  or  differential 
pulley.  94.  Center  of  gravity.  95.  Center  of  mass.  96.  Stable, 
unstable  and  neutral  equilibrium.  97.  Weighing  machines. 


CONTENTS  ix 

PART  II 
PROPERTIES  OF  MATTER 

CHAPTER  VIII 

PAGE 
THE  THREE  STATES  OF  MATTER  AND  THE  GENERAL  PROPERTIES  OP 

MATTER 137 

Section  98.  The  three  states  of  matter.  99.  Structure  of  matter. 
100.  Conservation  of  matter.  101.  General  properties  of  matter. 
102.  Intermolecular  attraction  and  the  phenomena  to  which  it 
gives  rise.  103.  Elasticity,  general  discussion. 

CHAPTER  IX 

PROPERTIES  OF  SOLIDS 144 

Section  104.  Properties  enumerated  and  defined.  105.  Elasticity, 
elastic  limit  and  elastic  fatigue  of  solids.  106.  Tensile  stress,  and 
tensile  strain.  107.  Hooke's  law  and  Young's  modulus.  108. 
Yield  point,  tensile  strength,  breaking  stress.  109.  Strength  of 
horizontal  beams.  110.  Three  kinds  of  elasticity  of  stress  and  of 
strain;  and  the  three  moduli.  111.  The  rigidity  of  a  shaft  and  the 
power  transmitted. 

CHAPTER  X 

THE  PROPERTIES  OF  LIQUIDS  AT  REST 155 

Section  112.  Brief  mention  of  properties.  113.  Hydrostatic  pres- 
sure. 114.  Transmission  of  pressure.  115.  The  Hydrostatic 
paradox.  116.  Relative  densities  of  liquids  by  balanced  columns. 
117.  Buoyant  force.  118.  The  principle  of  Archimedes.  119. 
Immersed  floating  bodies.  120.  Application  of  Archimedes' 
principle  to  bodies  floating  upon  the  surface.  121.  Center  of 
buoyancy.  122.  Specific  gravity.  123.  The  hydrometer.  124. 
Surf  ace  tension.  125.  Surface  a  minimum.  126.  Numerical  value 
of  surface  tension.  127.  Effect  of  impurities  on  surface  tension  of 
water.  128.  Capillarity.  129.  Capillary  rise  in  tubes,  wicks,  and 
soil.  130.  Determination  of  surface  tension  from  capillary  rise  in 
tubes. 

CHAPTER  XI 

PROPERTIES  OF  GASES  AT  REST 177 

Section  131.  Brief  mention  of  properties.  132.  The  earth's  atmos- 
phere. 133.  Height  of  the  atmosphere.  134.  Buoyant  effect, 
Archimedes'  principle,  lifting  capacity  of  balloons.  135.  Pressure 
of  the  atmosphere.  136.  The  mercury  barometer.  137.  The 
aneroid  barometer.  138.  Uses  of  the  barometer.  139.  Boyle's 
law.  140.  Boyle's  law  tube,  isothermals  of  a  gas.  141.  The 
manometers  and  the  Bourdon  gage. 


x  CONTENTS 

CHAPTER  XII 

PAGE 

PROPERTIES  OF  FLUIDS  IN  MOTION 194 

Section  142.  General  discussion.  143.  Gravity  flow  of  liquids. 
144.  The  siphon.  145.  The  suction  pump.  146.  The  force  pump. 
147.  The  mechanical  air  pump.  148.  The  Sprengel  mercury 
pump.  149.  The  windmill  and  the  electric  fan.  150.  Rotary 
blowers  and  rotary  pumps.  151.  The  turbine  water  wheel.  152. 
Pascal's  law.  153.  The  hydraulic  press.  154.  The  hydraulic 
elevator.  155.  The  hydraulic  ram.  156.  Diminution  of  pressure 
in  regions  of  high  velocity.  157.  The  injector.  158.  Ball  and 
jet.  159.  The  curving  of  a  baseball. 

PART  III 
HEAT 

CHAPTER  XIII 

THERMOMETRY  AND  EXPANSION 217 

Section  160.  The  nature  of  heat.  161.  Sources  of  heat.  162. 
Effects  of  heat.  163.  Temperature.  164.  Thermometers.  165. 
The  mercury  thermometer.  166.  Thermometer  scales.  167. 
Other  thermometers.  168.  Linear  expansion.  169.  Coefficient 
of  linear  expansion.  170.  Practical  applications  of  equalities  and 
differences  in  coefficient  of  linear  expansion.  171.  Cubical  expan- 
sion; Charles's  law.  172.  The  absolute  temperature  scale.  173. 
The  general  law  of  gases.  174.  The  thermocouple  and  the 
thermopile. 

CHAPTER  XIV 

HEAT  MEASUREMENT,  OR  CALORIMETRY 243 

Section  175.  Heat  units.  176.  Thermal  capacity.  177.  Specific 
heat.  178.  The  two  specific  heats  of  a  gas.  179.  The  law  of 
Dulong  and  Petit.  180.  Specific  heat,  method  of  mixtures.  181. 
Heat  of  combustion.  182.  Heat  of  fusion  and  heat  of  vaporiza- 
tion. 183.  Bunsen's  ice  calorimeter.  184.  The  steam  calorimeter. 
185.  Importance  of  the  peculiar  heat  properties  of  water.  186. 
Fusion  and  melting  point.  187.  Volume  change  during  fusion. 
188.  Regelation.  189.  Glaciers.  190.  The  ice  cream  freezer. 

CHAPTER  XV 

VAPORIZATION 260 

Section  191.  Vaporization  defined.  192.  Evaporation  and  ebulli- 
tion. 193.  Boiling  point.  194.  Effect  of  pressure  on  the  boiling 
point.  195.  Geysers.  196.  Properties  of  saturated  vapor. 
197.  Cooling  effect  of  evaporation.  198.  Wet-and-dry-bulb 
hygrometer.  199.  Cooling  effect  due  to  evaporation  of  liquid 


CONTENTS  xi 

PAGE 

carbon  dioxide.  200.  Refrigeration  and  ice  manufacture  by 
the  ammonia  process.  201.  Critical  temperature  and  critical 
pressure.  202.  Isothermals  for  carbon  dioxide.  203.  The  Joule- 
Thomson  experiment.  204.  Liquefaction  of  gases.  205.  The 
cascade  method  of  liquefying  gases.  206.  The  regenerative 
method  of  liquefying  gases. 

CHAPTER  XVI 

TRANSFER    OF    HEAT 283 

Section  207.  Three  methods  of  transferring  heat.  208.  Convec- 
tion. 209.  Conduction.  210.  Thermal  conductivity.  211.  Wave 
motion,  wave  length  and  wave  velocity.  212.  Interference  of 
wave  trains.  213.  Reflection  and  refraction  of  waves.  214. 
Radiation.  215.  Factors  in  heat  radiation.  216.  Radiation  and 
absorption.  217.  Measurement  of  heat  radiation.  218.  Trans- 
mission of  heat  radiation  through  glass,  etc.  219.  The  general 
case  of  heat  radiation  striking  a  body. 

CHAPTER  XVII 

METEOROLOGY 302 

Section  220.  General  discussion.  221.  Moisture  in  the  atmos- 
phere. 222.  Hygrometry  and  -hygrometers.  223.  Winds,  trade 
winds.  224.  Land  and  sea  breezes.  225.  Cyclones.  226.  Tor- 
nadoes. 

CHAPTER  XVIII 

STEAM  ENGINES  AND  GAS  ENGINES 311 

Section  227.  Work  obtained  from  heat — thermodynamics.  228. 
Efficiency.  229.  The  steam  engine.  230.  Condensing  engines. 
231.  Expansive  use  of  steam,  cut-off  point.  232.  Power.  233. 
The  slide  valve  mechanism.  234.  The  indicator.  235.  The  steam 
turbine.  236.  Carnot's  cycle.  237.  The  gas  engine — fuel,  carbu- 
retor, ignition  and  governor.  238.  Multiple-cylinder  engines. 
239.  The  four-cycle  engine.  240.  The  two-cycle  engine. 

INDEX    .  .  335 


PART  I 
MECHANICS 


\> 


. 


_\J 

< 


MECHANICS  AND  HEAT 

CHAPTER  I 
MEASUREMENT 

1.  The  Three  Fundamental  Quantities. — The  measurement 
of  physical  quantities  is  absolutely  essential  to  an  exact  and  scien- 
tific study  of  almost  any  physical  phenomenon.  For  this  reason, 
Measurement  is  usually  the  topic  first  discussed  in  a  course  in 
Physics.  The  popular  expressions,  "quite  a  distance,"  a  "large 
quantity,"  etc.,  are  too  indefinite  to  satisfy  the  scientific  mind. 
A  physical  quantity  may  be  defined  as  anything  that  can  be 
measured.  The  measurement  of  length,  mass,  and  time  are  of 
special  importance  and  will  therefore  be  considered  first. 
Indeed,  almost  all  physical  quantities  may  be  expressed  in 
terms  of  one  or  more  of  these  three  quantities,  for  which  reason 
they  are  called  the  fundamental  quantities.  In  the  case  of  some 
physical  quantities  this  is  at  once  apparent.  Thus,  to  measure 
the  area  of  a  piece  of  land,  it  is,  as  a  rule,  only  necessary  to 
measure  the  distance  across  it  north  and  south  (say  LI)  and  then 
east  and  west  (L2).  The  product  of  these  two  dimensions,  Z/iL2, 
is  then  an  area.  If  it  is  required  to  find  how  many  "yards"  of 
earth  have  been  removed  in  digging  a  cellar,  not  only  the  length 
and  width  must  be  known,  but  also  the  depth  (L3).  The  result 
evidently  involves  a  length  (i.e.,  distance)  only,  since  volume  = 
LiL2L3.  Coal,  grain,  etc.,  are  measured  in  terms  of  mass.  If 
the  quantity  involved  is  the  time  between  two  dates  it  is,  of 
course,  measured  in  terms  of  time.  If  a  train  goes  from  one  city 
to  another  in  a  known  time  T,  its  average  velocity  is  the  distance 
between  the  two  points  (i.e.,  a  length)  divided  by  the  time 
required  to  traverse  that  distance,  or 

Velocity  =| 

A  force  may  be  measured  in  terms  of  the  rate  at  which  it 
changes  the  velocity  of  a  body  of  known  mass  upon  which  it  acts. 
Velocity,  as  we  have  just  seen,  is  a  quantity  involving  both 

1 


2  MECHANICS  AND  HEAT 

length  and  time;  hence,  force  must  be  a  quantity  involving  all 
three  fundamental  quantities.  In  like  manner  it  may  be  shown 
that  other  physical  quantities,  e.g.,  power,  work,  electric 
charge,  electric  current,  etc.,  are  expressible  in  terms  of  one  or 
more  of  the  three  fundamental  quantities — length,  mass,  and 
time. 

2.  Units  and  Numerics. — In  order  to  measure  and  record 
the  value  of  any  quantity,  it  is  necessary  to  have  a  unit  of  that 
same  quantity  in  which  to  express  the  result.     Thus  if  we  meas- 
ure the  length  of  a  board  with  a  foot  rule  and  find  that  we  must 
apply  it  ten  times,  and  that  the  remainder  is  then  half  the  length 
of  the  rule,  we  say  that  the  length  of  the  board  is  10^  ft.     If 
this  same  board  is  measured  with  a  yard  stick,  3|  yds.  is  the 
result;  while,  if  the  inch  is  the  unit,  126  inches  is  the  result. 
Here  the  foot,  the  yard,  or  the  inch  is  the  Unit,  and  the  10^, 
3^,   or  126  is  the  Numeric.     Evidently  the  larger  the  unit,  the 
smaller  the  numeric,  and  vice  versa.     Thus,    in   expressing    a 
weight  of  2  tons  as  4000  Ibs.,  the  numeric  becomes  2000  times 
as  large  because  the  unit  chosen  is  1/2000  as  large  as  before. 

3.  Fundamental  Units. — In  the  British  System  of  measure- 
ment, which  is  used  in  practical  work  in  the  United  States  and 
Great  Britain,  the  units  of  length,  mass,  and  time  are  respec- 
tively ihefoot,  the  pound,  and  the  second.     It  is  often  termed  the 
foot-pound-second  system,  or  briefly  the  "  F.P.S."  system.    Since, 
as  has  been  pointed  out,  nearly  all  physical  quantities  may  be 
expressed  in  terms  of  one  or  more  of  the  above  quantities,  the 
above  units  are  called  Fundamental  Units.     (The  fundamental 
units  of  the  metric  system  are  given  in  Sec.  5.) 

4.  Standards  of  Length,  Mass,  and  Time. — If  measurements 
made  now  are  to  be  properly  interpreted  several  hundred  years 
later,  it  is  evident  that  the  units  involved  must  not  be  subject 
to  change.     To  this  end  the  British  Government  has  had  made, 
and  keeps  at  London,  a  bronze  bar  having  near  each  end  a  fine 
transverse  scratch  on  a  gold  plug.     The  distance  between  these 
two  scratches,  when  the  temperature  of  the  bar  is  62°  Fahren- 
heit, is  the  standard  yard.     At  the  same  place  is  kept  a  piece  of 
platinum  of  1  Ib.  mass.     This  bar  and  this  piece  of  platinum 
are  termed   the   Standards   of  length  and   mass  respectively. 
The  standard  for  time  measurement  is  the  mean  solar  day,  and 
the  second  is  then  fixed  as  the  1/60X1/60X1/24,  or  1/86400 
part  of  a  mean  solar  day. 


*  ' 

MEASUREMENT  3 

The  Day — Sidereal,  Solar,  and  Mean  Solar. — Very  few  things  so  com- 
monplace as  the  day,  are  so  little  understood.  The  time  that  elapses 
between  two  successive  passages  of  a  star  (a  true  star,  not  a  planet) 
across  the  meridian  (a  north  and  south  line),  in  other  words  the  time 
interval  from  "star  noon"  to  "star  noon,"  is  a  Sidereal  Day.  From 
"sun  noon"  to  "sun  noon"  is  a  Solar  Day.  The  longest  solar  day  is 
nearly  a  minute  longer  than  the  shortest.  The  average  of  the  365  solar 
days  is  the  Mean  Solar  Day.  The  mean  solar  day  is  the  day  commonly 
used.  It  is  exactly  24  hours.  The  sidereal  day,  which  is  the  exact 
time  required  for  the  earth  to  make  one  revolution  on  its  axis,  is 
nearly  four  minutes  shorter  than  the  mean  solar  day. 

The  cause  for  the  four  minutes  difference  between  the  sidereal  day  and 
the  solar  day  may  be  indicated  by  two  or  three  homely  illustrations.  If 
a  silver  dollar  is  rolled  around  another  dollar,  without  slipping,  it  will  be 
found  that  the  moving  dollar  makes  two  rotations  about  its  axis, 
while  making  one  revolution  about  the  stationary  dollar.  The  moon 
always  keeps  the  same  side  toward  the  earth,  and  for  this  very  reason 
rotates  once  upon  its  axis  for  each  revolution  about  the  earth.  Compare 
constantly  facing  a  chair  while  you  walk  once  around  it.  You  will  find 
that  you  have  turned  around  (on  an  axis)  once  for  each  revolution  about 
the  chair.  If,  now,  you  turn  around  in  the  same  direction  as  before, 
three  times  per  revolution,  you  will  find  that  you  face  the  chair  but 
twice  per  revolution.  For  exactly  the  same  reason  the  earth  must 
rotate  366  times  on  its  axis  during  one  revolution  about  the  sun,  in  order 
to  "face"  the  sun  365  times.  Consequently  the  sidereal  day  is,  using 
round  numbers,  365/366  as  long  as  the  mean  solar  day,  or  about  four 
minutes  shorter. 

Variation  in  the  Solar  Day. — If  the  orbit  of  the  earth  around  the  sun 
were  an  exact  circle,  and  it,  further,  the  axis  of  rotation  of  the  earth 
were  at  right  angles  to  the  plane  of  its  orbit  (plane  of  the  ecliptic),  then 
all  solar  days  would  be  of  equal  length.  The  orbit,  however,  is  slightly 
elliptical,  the  earth  being  nearer  to  the  sun  in  winter  and  farther  from  it 
in  summer  than  at  other  seasons;  and  the  axis  of  the  earth  lacks  23°. 5 
of  being  at  right  angles  to  the  plane  of  the  ecliptic. 

Let  S,  (Fig.  1)  represent  the  sun,  E,  the  earth  on  a  certain  day,  and  E', 
the  earth  a  sidereal  day  later  (distance  EE'  is  exaggerated).  Let  the 
curved  arrow  indicate  the  rotary  motion  of  the  earth  and  the  straight 
arrow,  the  motion  in  its  orbit.  When  the  earth  is  at  E,  it  is  noon 
at  point  A;  i.e.,  AS  is  vertical;  while  at  E',  the  earth  having  made 
exactly  one  revolution,  the  vertical  at  A  is  AB,  and  it  will  not  be  noon 
until  the  vertical  (hence  the  earth)  rotates  through  the  angle  0.  This 
requires  about  four  minutes  (0  being  much  smaller  than  drawn),  causing 
the  solar  day  to  be  about  four  minutes  longer  than  the  sidereal  day. 
The  stars  are  so  distant  that  if  AS  points  toward  a  star,  then  AB, 
which  is  parallel  to  it,  points  at  the  same  star  so  far  as  the  eye  can  detect. 


4  MECHANICS  AND  HEAT 

Hence  the  sidereal  day  gives,  as  above  stated,  the  exact  time  of  one 
revolution  of  the  earth. 

When  the  earth  is  nearest  to  the  sun  (in  December)  it  travels  fastest; 
i.e.,  when  AS  is  shortest,  EE'  is  longest.  Obviously  both  of  these 
changes  increase  0  and  hence  make  the  solar  day  longer.  The  effect 
of  the  above  23°.5  angle,  in  other  words,  the  effect  due  to  the 
obliquity  of  the  earth's  axis,  is  best  explained  by  use  of  a  model. 
We  may  simply  state,  however,  that  due  to  this  cause  the  solar  day  in 
December  is  still  further  lengthened.  As  a  result  it  is  nearly  a  minute 
longer  than  the  shortest  solar  day,  which  is  in  September. 

When  the  solar  days  are  longer  than  the  mean  solar  day  (24  hour  day) 
the  sun  crosses  the  meridian,  i.e.,  "transit"  occurs, 
later  and  later  each  day;  while  when  they  are 
shorter,  the  transit  occurs  earlier  each  day.  In 
February,  transit  occurs  at  about  12:15  mean  so- 
lar  time  (i.e.,  clock  time),  at  which  date  the  alma- 
nac  records  sun  "slow"  15  minutes.  In  early 
November  the  sun  is  about  15  minutes  "fast." 
These  are  the  two  extremes. 

6.  The  Metric  System. — This  system  is  in  common  use  in 
most  civilized  countries  except  the  United  States  and  Great 
Britain,  while  its  scientific  use  is  universal.  The  fundamental 
units  of  the  Metric  System  of  measurement  are  the  centimeter, 
the  gram,  and  the  second.  It  is  accordingly  called  the  centimeter- 
gram-second  system,  or  briefly  the  "C.G.S."  system.  This 
system  far  surpasses  the  British  system  in  simplicity  and  facility 
in  computation,  because  its  different  units  for  the  measurement 
of  the  same  quantity  are  related  by  a  ratio  of  10,  or  10  to  some 
integral  power,  as  100,  1000,  etc.  The  centimeter  (cm.)  is  the 
1/100  part  of  the  length  of  a  certain  platinum-iridium  bar  when  at 
the  temperature  of  melting  ice.  This  bar,  whose  length  (between 
transverse  scratches,  at  0°C.)  is  1  meter  (m.),  is  kept  at  Paris  by 
the  French  Government.  The  gram  is  the  1/1000  part  of  the  mass 
of  a  certain  piece  of  platinum  (the  standard  kilogram)  kept  at  the 
same  place.  The  milligram  is  1/1000  gm.,  and  the  millimeter 
(mm.)  is  1/1000  meter.  The  second  is  the  same  as  in  the  British 
system.  The  above  meter  bar  and  kilogram  mass  are  respectively 
the  Standards  of  length  and  mass  in  the  Metric  System. 

6.  Conversion  of  Units. — In  this  course  both  systems  of 
units  will  be  used,  because  both  are  frequently  met  in  general 
reading.  Some  practice  will  also  be  given  in  converting  results 
expressed  in  terms  of  the  units  of  one  system  into  units  of  the 


MEASUREMENT  5 

other  (see  problems  at  the  close  of  this  chapter).  To  do  this  it 
is  only  necessary  to  know  that  1  inch  =  2.54  cm.  and  1  kilogram 
(  =  1000  gm.)  =  2.2046  Ibs.,  or  approximately  2.2  Ibs.  These  two 
ratios  should  be  memorized,  and  perhaps  also  the  fact  that  the 
meter  =  39.37  in.  From  the  first  ratio  it  will  be  seen  that  the 
numeric  is  made  30.48  (or  12X2.54)  times  as  large  whenever  a 
certain  length  is  expressed  in  centimeters  instead  of  in  feet.  The 
relation  between  all  other  units  in  the  two  systems  can  readily 
be  obtained  if  the  above  two  ratios  are  known. 

7.  Measurement    of    Length.  —  The    method    employed    in 
measuring  the  length  of  any  object  or  the  distance  between  any 
two  points,  will  depend  upon  the  magnitude  of  the  distance  to 
be  measured,  and  the  accuracy  with  which  the  result  must  be 
determined.     For  many  purposes,  either  the  meter  stick  or  the 
foot  rule  answers  very  well;  while  for  other  purposes,  such  as 
the  measurement  of  the  thickness  of  a  sheet  of  paper,  both  are 
obviously  useless.     For  more  accurate  measurements,  several 
instruments  are  in  use,  prominent  among  which  are  the  vernier 
caliper,  the  micrometer  caliper,  and  the  micrometer  microscope. 

8.  The   Vernier   Caliper.  —  In  Fig.   2  is  shown  a  simplified 
form  of  the  vernier  caliper  from  which  the  important  principle 
of  the  vernier  may  be  readily  comprehended.     This  vernier  cali- 


r 
n 

L       J 

\ 

£> 

.  ... 

-F 

1°,      ,   i   !5,Ci   ,   ,  fl    0 

C 

Ul          1     1     1     1     1     I     I          f    |     1     1     1     1     1         7 

a      5          ,4    Jo              hfe                 do      \ 

$ 


FIG.  2. 

per  consists  of  a  bar  A,  having  marked  near  one  edge  a  scale  in 
millimeter  divisions  Rigidly  attached  to  A  is  the  jaw  B, 
whose  face  F  is  accurately  perpendicular  to  A,  and  parallel  to 
the  face  of  jaw  D,  attached  to  bar  C.  C  may  be  slid  along  A 
until  D  strikes  B,  if  there  is  nothing  between  the  jaws.  While 
in  this  position,  a  scale  of  equal  divisions  is  ruled  upon  C  having 
its  zero  line  in  coincidence  with  the  zero  line  of  A,  and  its  tenth 
line  in  coincidence  with  the  ninth  line  on  A.  The  scale  on  C  is 
called  the  vernier  scale  and  that  on  A,  the  main  scale.  Obvi- 
ously, the  vernier  divisions  are  1/10  mm.  shorter  than  the  main 


6  MECHANICS  AND  HEAT 

scale  divisions;  i.e.,  they  are  9/10  as  long,  since  10  vernier  divi- 
sions just  equal  9  scale  divisions. 

To  measure  the  length  of  the  block  E,  place  it  between  the 
jaws  D  and  B,  as  shown.  Since  the  two  zero  lines  coincide  when 
the  jaws  are  together,  the  length  of  the  block  must  be  equal  to 
the  distance  between  the  two  zeros,  or  3  mm.,  plus  the  small  dis- 
tance a.  But  if  line  2  on  the  vernier  coincides  with  a  line  on  the 
main  scale,  as  shown,  then  a  is  simply  the  difference  in  length 
between  2  vernier  divisions  and  2  main  scale  divisions,  or  0.2 
mm.  The  length  of  E  is  then  3.2  mm. 

If  C  were  slid  to  the  right  1/10  mm.,  line  3  on  the  vernier 
would  coincide  with  a  main  scale  line,  and  a  would  then  equal  0.3 
mm.;  so  that  the  distance  between  the  jaws  would  be  3.3  mm. 
Evidently,  the  above  1/10  mm.  is  the  least  motion  of  C  that  can 
be  directly  measured  by  the  vernier.  This  distance  (1/10  mm.) 
is  called  the  sensitiveness  of  this  vernier.  If  the  divisions  on  A 
had  been  made  1/20  inch,  and  25  vernier  divisions  had  been 
made  equal  to  24  main  scale  divisions,  then  the  sensitiveness  or 
difference  between  the  length  of  a  main  scale  division  and  a  ver- 
nier division  would  be  1/500  inch.  For  the  vernier  divisions, 
being  1/25  division  shorter  than  the  main  scale  divisions  (i.e., 
24/25  as  long),  are  1/25X1/20  or  1/500  inch  shorter. 

This  arrangement  of  two  scales  of  slightly  different  spacing, 
free  to  slide  past  each  other,  is  an  application  of  the  Vernier 
Principle.  This  principle  is  much  employed  in  making  measur- 
ing instruments.  Instead  of  having  10  vernier  spaces  equal  to 
9  spaces  on  the  main  scale,  the  ratio  may  be  25  to  24  as  men- 
tioned, or  50  to  49,  16  to  15,  etc.,  according  to  the  use  that  is  to 
be  made  of  the  instrument.  In  the  case  of  circular  verniers 
and  scales  on  surveying  instruments,  the  above-mentioned  ratio 
is  usually  30  to  29  or  else  60  to  59,  because  they  are  to  be  read  in 
degrees,  minutes,  and  seconds  of  arc.  If  the  vernier  principle  is 
thoroughly  understood,  there  should  be  no  difficulty  in  reading 
any  vernier,  whether  straight  or  circular,  in  which  a  convenient 
ratio  is  employed. 

9.  The  Micrometer  Caliper. — The  micrometer  caliper  (Fig. 
3)  consists  of  a  metal  yoke  A,  a  stop  S,  a  screw  B  whose  threads 
fit  accurately  the  threads  cut  in  the  hole  through  A,  and  a  sleeve 
C  rigidly  connected  to  B.  When  B  and  S  are  in  contact,  the 
edge  E  of  C  is  at  the  zero  of  scale  D;  consequently  the  dis- 
tance from  S  to  B,  in  other  words  the  thickness  of  the  block  F 


MEASUREMENT  7 

as  sketched,  is  equal  to  the  distance  from  this  zero  to  E.  If 
the  figure  represents  the  very  common  form  of  micrometer  cali- 
per  in  which  the  "pitch"  of  B  (i.e.,  the  distance  B  advances  for 
each  revolution)  is  1/2  mm.,  D  is  a  scale  of  millimeter  divi- 
sions, and  the  circumference  of  C  at  E  is  divided  into  50  equal 
divisions;  then  the  thickness  of  F  is  4.5  mm.  plus  the  slight  dis- 
tance that  B  moves  when  E  turns  through  6  of  its  divisions, 
or  6/50  of  a  revolution.  But  6/50X1/2  mm.  =  0.06  mm.;  so 
that  the  thickness  of  E  is  4.5+0.06  or  4.56  mm.  It  should  be 
explained  that  if  the  instrument  is  properly  adjusted,  then,  when 
B  and  S  are  in  contact,  the  zero  of  E  and  the  zero  of  D  coincide. 
Accordingly  if  the  zero  of  E  were  exactly  in  line  with  scale  D, 
then  4.5  would  be  the  result.  As  sketched,  however,  it  is  6/50 
of  a  revolution  past  the  position  of  alignment  with  D,  which 


FIG.  3. 

adds  0.06  mm.  to  the  distance  between  B  and  S  as  already 
shown. 

If  C  were  turned  in  the  direction  of  arrow  a  through  1/50  revo- 
lution, then  line  7  of  E,  instead  of  line  6,  would  come  in  line 
with  D,  and  B  would  have  moved  1/50X1/2  mm.,  or  0.01  mm. 
farther  from  S.  This,  the  least  change  in  setting  that  can  be 
read  directly  without  estimating,  is  called  the  Sensitiveness  of 
an  instrument  (see  Sec.  8).  Thus  the  sensitiveness  of  this 
micrometer  caliper  is  0.01  mm. 

10.  The  Micrometer  Microscope. — The  micrometer  micro- 
scope consists  of  an  ordinary  compound  microscope,  having 
movable  crosshairs  in  the  barrel  of  the  instrument  where  the 
magnified  image  of  the  object  to  be  measured  is  formed.  These 
crosshairs  may  be  moved  by  turning  a  micrometer  screw  similar 
to  B  in  Fig.  3. 

If  it  is  known  how  many  turns  are  required  to  cause  the  cross- 
hairs to  move  over  one  space  of  a  millimeter  scale,  placed  on 


8  MECHANICS  AND  HEAT 

the  stage  of  the  microscope,  and  also  what  part  of  a  turn  will 
cause  them  to  move  the  width  of  a  small  object  also  placed  on 
the  stage,  the  diameter  of  the  object  can  be  at  once  calculated. 

11.  Measurement  of  Mass,  Inertia. — Consider  two  large  pieces 
of  iron,  provided  with  suitable  handles  for  seizing  them,  each  one 
resting  upon  a  light  and  nearly  frictionless  truck  on  a  level  steel 
track,  and  hence  capable  of  being  moved  in  a  horizontal  direc- 
tion with  great  freedom.  If  a  person  is  brought  blindfolded  and 
permitted  to  touch  only  the  handles,  he  can  very  quickly  tell 
by  jerking  them  to  and  fro  horizontally,  which  one  contains  the 
greater  amount  of  iron.  If  one  piece  of  iron  is  removed  and 
replaced  by  a  piece  of  wood  of  the  same  size  as  the  remaining 
piece  of  iron,  he  would  immediately  detect  that  the  piece  of 
wood  moved  more  easily  and  would  perhaps  think  it  to  be  a  very 
small  piece  of  iron.  The  difference  which  he  detects  is  certainly 
not  difference  in  volume,  as  he  is  not  permitted  either  to  see  or 
to  feel  them;  neither  is  it  difference  in  weight,  since  he  does  not 
lift  them.  It  is  difference  in  Mass  that  he  detects.  Hence 
Mass  may  be  denned  as  that  property  of  matter  by  virtue  of  which 
it  resists  being  suddenly  set  into  motion,  or,  if  already  in  motion, 
resists  being  suddenly  brought  to  rest. 

Inertia  and  Mass  are  synonymous;  inertia  being  used  in  a 
general  way  only,  while  mass  is  used  in  a  general,  qualitative 
way  and  also  in  a  quantitative  way.  Thus  we  speak  of  a  large 
mass,  great  inertia,  a  5-lb.  mass,  etc.,  but  not  of  5  Ibs.  inertia. 

If  it  were  possible,  by  the  above  method,  for  the  person  to 
make  accurate  determinations,  and  if  he  found  that  one  piece 
had  just  twice  as  much  mass  as  the  other,  then  upon  weighing 
them  it  would  be  found  that  one  piece  was  exactly  twice  as 
heavy  as  the  other.  In  other  words,  the  Weight  of  any  body  is 
proportional  to  its  Mass.  The  weight  of  a  body  is  simply  the 
attractive  pull  of  the  earth  upon  it;  hence  we  see  that  the  pull  of 
the  earth  upon  any  body  depends  upon  the  mass  of  the  body,  and 
therefore  affords  a  very  convenient,  and  also  very  accurate 
means  of  comparing  masses. 

Thus  the  druggist,  using  a  simple  beam  balance,  "weighs 
out"  a  pound  mass  of  any  chemical  by  placing  a  standard  pound 
mass  in  one  pan  and  then  pouring  enough  of  the  chemical  into 
the  other  pan  to  exactly  "balance"  it.  That  is,  the  amount  of 
chemical  in  one  pan  is  varied  until  the  pull  of  the  earth  on  the 
chemical  at  one  end  of  the  beam  is  made  exactly  equal  to  the 


MEASUREMENT  9 

pull  of  the  earth  on  the  standard  pound  mass  at  the  other  end. 
He  then  knows,  since  the  pull  of  the  earth  on  each  is  equal, 
that  their  weights,  and  consequently  their  masses,  are  equal. 
Weights,  and  hence  masses,  may  be  compared  also  by  means 
of  the  steel-yard,  the  spring  balance,  and  the  platform  scale. 
These  devices  will  be  discussed  later  in  the  course. 

The  mass  of  a  body  is  absolutely  constant  wherever  it  is 
determined,  while  its  weight  becomes  very  slightly  less  as  it  is 
taken  up  a  mountain  or  taken  toward  the  equator.  This  is 
due  partly  to  the  fact  that  the  body  is  slightly  farther  from  the 
earth's  center  at  those  points,  and  partly  to  the  rotary  motion  of 
the  earth  (see  centrifugal  force,  Sec.  58).  The  polar  diameter  of 
the  earth  is  about  27  miles  less  than  its  equatorial  diameter. 
A  given  object  weighed  at  St.  Louis  and  then  at  St.  Paul  with 
the  same  spring  balance  should  show  an  increase  in  weight  at 
the  latter  place;  whereas  if  weighed  with  the  same  beam  balance 
at  both  places,  there  should  be  no  difference  in  the  weights 
read.  The  weight  of  the  object  actually  does  increase,  but  the 
weight  of  the  counterbalancing  standard  masses  used  with  the 
beam  balance  also  increases  in  the  same  proportion. 

12.  Measurement  of  Time. — A  modern  instrument  for 
measuring  time  must  have  these  three  essentials:  (1)  a  device1 
for  measuring  equal  intervals  of  time,  i.e.,  for  time  "spacing," 
(2)  a  driving  mechanism,  (3)  a  recording  mechanism.  In 
the  case  of  the  clock,  (1)  is  the  pendulum,  (2)  is  the  mainspring  or 
weights,  train  of  wheels  and  escapement,  and  (3)  is  the  train  of 
wheels  and  the  hands.  In  the  watch,  the  balance  wheel  and 
hairspring  take  the  place  of  the  pendulum. 

The  necessity  for  the  pendulum  or  its  equivalent,  and  the 
recording  mechanism,  is  obvious.  Friction  makes  the  driving 
mechanism  necessary.  The  escapement  clutch  attached  to  the 
pendulum  is  shaped  with  such  a  slant  that  each  time  it  releases 
a  cog  of  the  escapement  wheel  it  receives  from  that  wheel  a 
slight  thrust  just  sufficient  to  compensate  for  friction,  which 
would  otherwise  soon  bring  the  pendulum  to  rest.  If  the 
pendulum,  as  it  vibrates,  releases  a  cog  each  second,  and  if 
the  escapement  wheel  has  20  cogs,  the  latter  will,  of  course, 
make  a  revolution  in  20  seconds.  It  is  then  an  easy  matter  to 
design  a  connecting  train  of  geared  wheels  and  pinions  between 
it  and  the  post  to  which  the  minute  hand  is  attached,  so  that 
the  latter  will  make  one  revolution  in  an  hour.  In  the  same 


10  MECHANICS  AND  HEAT 

way  the  hour  hand  is  caused  to  make  one  revolution  in  twelve 
hours. 

In  the  hourglass  of  olden  times,  and  in  the  similar  device,  the 
clepsydra  or  water  dropper  of  the  Ancient  Greeks,  only  the  time 
"spacing"  is  automatic.  The  observer  became  the  driving 
mechanism  by  inverting  the  hourglass  at  the  proper  moment; 
and  by  either  remembering  or  recording  how  many  times  he  had 
inverted  it,  he  became  also  the  recording  mechanism. 

Other  time  measurers,  in  which  only  time  spacing  is  present, 
are  the  earth  and  the  moon.  The  rotation  of  the  earth  about 
its  axis  determines  our  day,  while  its  revolution  about  the  sun 
determines  our  year.  The  revolution  of  the  moon  about  the 
earth  determines  our  lunar  month,  which  is  about  28  days. 

PROBLEMS 
jr. 

1.  What  is  the  height  in  feet  and  inches  of  a  man  who  is  1  m.  80  cm.  tall? 
Reduce  5  ft.  4.5  in.  to  centimeters. 

2.  What  does  a  160-lb.  man  weigh  in  grams?     In  kilograms?     Reduce 
44  kilograms  240  grams  to  pounds. 

3.  Reduce  100  yds.  to  meters.     What  part  of  a  mile  is  the  kilometer? 

4.  A  cubic  centimeter  of  gold  weighs  19.3  gm.     Find  the  weight  of  1  cu. 
ft.  of  gold  in  grams.     In  pounds. 

5.  One  cm.3  of  glycerine  weighs  1.27  gms.     How  many  pounds  will  1 
gaUon  (231  in.3)  weigh? 

6.  If  a  man  can  run  100  yds.  in  10  sec.,  how  long  will  he  require  for  the 
100  meter  dash?     Assume  the  same  average  velocity  for  both. 

7.  If,  in  Fig.  2,  the  main  scale  divisions  were  1/16  inch,  and  20  vernier 
divisions  were  equal  to  19  divisions  on  the  main  scale,  other  conditions  being 
as  shown,  what  would  be  the  length  of  El 

8.  The  pitch  of  a  certain  micrometer  caliper  is  1/20  inch  and  the  screw 
head  has  25  divisions.     After  setting  upon  a  block  and  then  removing  it, 
7  complete  turns  and  4  divisions  are  required  to  cause  the  screw  to  advance 
to  the  stop.     What  is  the  thickness  of  the  block? 

9.  Between  the  jaws  of  a  vernier  caliper  (Fig.  2)  is  placed  a  block  of  such 
length  that  line  5  of  the  vernier  scale  coincides  with  line  10  of  the  main  scale, 
and  consequently  the  zero  of  the  vernier  scale  is  a  short  distance  to  the  right 
of  line  5  of  the  main  scale.     If  the  main  scale  divisions  are  1/2  mm.,  and  25 
vernier  divisions  are  equal  in  length  to  24  main  scale  divisions,  what  is  the 
length  of  the  block? 

10.  What  is  the  sensitiveness  (see  Sec.  8)  of  the  vernier  caliper  in  problem 
7?     In  problem  9?     What  is  the  sensitiveness  of  the  micrometer  caliper  in 
problem  8? 


* 


CHAPTER  II 

<\  V  r 

VECTORS 

13.  Scalars  and  Vectors  Defined.  —  All  physical  quantities 
may  be  divided  into  two  general  c\assesf^Scalars  and  Vectors. 
A  scalar  quantity  is  one  that  is  fully  specified  if  its  magnitude 
only  is  given;  while  to  specify  a  vector  quantity  completely, 
not  only  its  magnitude,  but  also  its  direction  must  be  given. 
Hence  vectors  might  be  called  directed  quantities. 

Such  quantities  as  volume,  mass,  work,  energy,  and  quantity 
of  heat  or  of  electricity,  do  not  have  associated  with  them  any 
idea  of  direction,  and  are  therefore  scalars.  Force,  pressure, 
and  velocity,  must  have  direction  as  well  as  magnitude  given  or 
they  are  not  completely  specified;  therefore  they  are  vectors. 
Thus,  if  the  statement  is  made  that  a  certain  ship  left  port  at  a 
speed  of  20  miles  per  hour,  the  motion  of  the  ship  is  not  fully 
known.  The  statement  that  the  ship's  velocity  was  20  miles  an 
hour  due  north,  completely  specifies  the  motion  of  the  ship,  and 
conveys  the  full  meaning  of  velocity.  This  distinction  between 
speed  and  velocity  is  not  always  observed  in  popular  language, 
but  it  must  be  observed  in  technical  work. 

If  two  forces  FI  and  F2  act  upon  a  body,  say  a  boat  in  still 
water,  they  will  produce  no  effect,  if  equal  and  opposed;  i.e., 
if  the  angle  between  the  two  forces  is  180°.  If  this  angle  is  zero, 
i.e.,  if  both  forces  act  in  the  same  direction,  their  Resultant  F 
(Sec.  15),  or  the  single  force  that  would  produce  the  same  effect 
upon  the  boat  as  both  FI  and  Fz,  is  simply  their  sum,  or 

F  =  F1+FZ  (1) 

If  FI  is  greater  than  FZ,  then  when  the  angle  between  them  is 
180°,  that  is  when  ^i  and  F2  are  oppositely  directed,  we  have 

F=F,-F2  (2) 

The  resultant  F  has  in  Eq.  1  its  maximum  value,  and  in  Eq.  2 
its  minimum  value.  It  may  have  any  value  varying  between 
these  limits,  as  the  angle  between  F\  and  F2  varies  from  zero  to 
180°. 

o        { 


12  MECHANICS  AND  HEAT 

In  contrast  with  the  above  statements,  observe  that  in  scalar 
addition  the  result  is  always  simply  the  arithmetical  sum. 
Thus,  15  qts.  and  10  qts.  are  25  qts.;  while  the  resultant  of  a 
15-lb.  pull  and  a  10-lb.  pull  may  have  any  value  between  5  Ibs. 
and  25  Ibs.  and  it  may  also  have  any  direction,  depending  upon 
the  directions  of  the  two  pulls. 

Note  that  such  physical  objects  as  a  stone  or  a  train  are  neither 
scalars  nor  vectors.  Several  physical  quantities  relating  to  a 
stone  are  scalars;  viz.,  its  mass,  volume,  and  density;  while 
some  are  vectors;  viz.,  its  weight,  and,  if  in  motion,  its  velocity. 

14.  Representation  of  Vectors  by  Straight  Lines. — A  very 
simple  and  rapid  method  of  calculating  vectors,  called  the 
Graphical  Method,  depends  upon  the  fact  that  a  vector  may  be 
completely  represented  by  a  straight  line  having  at  one  end  an 
arrow  head.  Thus  to  represent  the  velocity  of  a  southwest 

" 


wind  blowing  at  the  rate  of  12  miles  an  hour,  a  line  (a)  2 
cm.  long,  or  (6)  4  cm.  long,  or  (c)  2  inches  long,  may  be  used  as 
shown  at  A,  Fig.  4.  In  case  (a),  1  cm.  represents  6  miles  an 
hour;  while  in  case  (6)  it  represents  3  mi.  an  hour.  In  case 
(c)  the  scale  is  chosen  the  same  as  in  case  (a),  except  that  lin., 
instead  of  1  cm.,  represents  6  miles  an  hour  velocity.  Any  con- 
venient scale  may  be  chosen.  In  each  case  the  length  of  the 
line  represents  the  magnitude  of  the  vector  quantity;  and  the 
direction  of  the  line  represents  the  direction  of  the  vector 
quantity. 

15.  Addition  of  Vectors,  Resultant. — The  vector  sum  or 
Resultant  (see  Sec.  13)  of  two  or  more  forces  or  other  vectors 
differs  in  general  from  either  the  arithmetical  or  the  algebraic 
sum.  By  the  Graphical  method,  it  may  be  found  as  follows. 
Choose  a  suitable  scale  and  represent  the  first  force  ^i  by  a  line 


VECTORS 


13 


having  an  arrow  head  as  shown  at  B,  Fig.  4.  Next,  from 
the  arrow  point  of  this  line,  draw  a  second  line  representing  the 
second  force  F2,  and  from  the  arrow  point  of  Fz  draw  a  line 
representing  Fs,  etc.  Finally  connect  the  beginning  of  the  first 
line  with  the  arrow  point  of  the  last  by  a  straight  line.  The 
length  of  this  line,  say  in  inches,  multiplied  by  the  number  of 
pounds  which  one  inch  represents  in  the  scale  chosen,  gives  the 


FIG.  5. 

magnitude  of  the  resultant  force  R.  The  direction  of  this  line 
gives  the  direction  of  the  resultant  force.  Obviously,  the  same 
scale  must  be  used  throughout.  An  example  involving  several 
velocities  will  further  illustrate  this  method  of  adding  vectors. 
Although  in  this  course  we  shall  apply  the  graphical  method  to 
only  force  and  velocity,  it  should  be  borne  in  mind  that  it  may 
be,  and  indeed  is,  applied  to  any  vector  quantity. 

A  steamboat,  which  travels  12  miles  an  hour  in  still  water,  is 


14  MECHANICS  AND  HEAT 

headed  due  east  across  a  stream  which  flows  south  at  the  rate  of 
5  miles  an  hour.  Let  us  find  the  velocity  of  the  steamboat. 
In  an  hour,  the  boat  would  move  eastward  a  distance  of  12 
miles  due  to  the  action  of  the  propeller,  even  if  the  river  did  not 
flow;  while  if  the  propeller  should  stop,  the  flow  of  the  river  alone 
would  cause  the  boat  to  drift  southward  5  miles  in  an  hour. 
Consequently,  if  subjected  to  the  action  of  both  propeller  and 
current  for  an  hour,  the  steamboat  would  be  both  12  miles  far- 
ther east  and  5  miles  farther  south,  or  at  D  (case  A,  Fig.  5).  By 
choosing  1  cm.  to  represent  4  miles  per  hr.,  the  " steam" 
velocity  would  be  represented  by  a  line  a,  3  cm.  in  length; 
while  the  "drift"  velocity  of  5  miles  an  hour  to  this  same 
scale,  would  be  represented  by  a  line  6,  1.25  cm.  in  length.  The 
length  (3.25  cm.)  of  the  line  OD  or  R  represents  the  magnitude 
of  the  steamboat's  velocity,  and  the  direction  of  this  line  gives 
the  course  of  the  boat,  or  the  direction  of  its  velocity.  The 
velocity  is  then  4X3.25  or  13  miles  an  hour  to  the  south  of  east 
by  an  angle  6  as  shown.  This  velocity  R,  of  13  miles  per  hour,  is 
the  resultant  or  vector  sum  of  the  two  velocities  a  and  6,  and  is 
evidently  the  actual  velocity  of  the  steamboat. 

By  the  analytical  method,  the  magnitude  of  the  resultant 
velocity  is  given  by  the  equation 


#  =  \/(12)2+(5)2 
and  its  direction  is  known  from  the  equation 

tan  5  =  5/12  =  0.417 
from  which  0  =  22°.38. 

If  the  steamboat  is  headed  southeast,  then  a\  and  &i  (case 
B,  Fig.  5)  represent  the  "steam"  and  "drift"  velocities  re- 
spectively, and  the  magnitude  of  the  resultant  velocity  R\, 
in  miles  per  hr.,  will  be  found  by  multiplying  the  length  of 
#1  in  centimeters  by  4.  If  the  analytical  method  is  employed, 
we  have  from  trigonometry, 

-Ri2  =  ai2+&i2+2a1&i  cos  0 

Suppose,  further,  that  it  is  required  to  find  the  actual  velocity 
of  a  man  who  is  walking  toward  the  right  side  of  the  steamboat 
at  the  rate  of  2  miles  an  hour,  when  the  boat  is  headed  as  shown 
in  case  B.  Let  a\,  b\,  and  Ci  represent  the  "steam,"  "drift," 
and  "walking"  velocities  respectively;  then  R2  represents  the 


VECTORS  15 

actual  velocity  of  the  man  as  shown  in  case  C,  Fig.  5.  If  the  man 
walks  toward  the  left  side  of  the  boat,  his  "walking"  velocity  is 
c2  and  his  actual  velocity  is  R3.  In  these  cases  his  velocity  could 
also  be  found  by  the  analytical  method,  but  not  so  readily. 

16.  The   Vector  Polygon. — In  cases  A  and  B  (Fig.  5),  the 
vector  triangle  is  used  in  determining  the  resultant;  while  in 
case  C,  the  vector  polygon,  whose  sides  are  a\,  bi,  c\  and  Rz,  is 

?  so  used.     In  general,  however,  many  vectors  are  involved,  the 
closing  side  of  the  polygon  represents  the  re- 
sultant of  all  the  other  vectors. 

If  a  man  were  to  run  toward  the  left  and 
rear  end  of  the  steamboat  in  the  direction  Rf 
at  the  speed  of  13  miles  per  hour  (case  A),  he 
would  appear  to  an  observer  on  shore  to  be 
standing  still  with  respect  to  the  shore. 
Hence  his  actual  velocity  is  zero.  Since  R' 
is  equal  to  R  and  oppositely  directed,  we  see 
that  the  three  vectors  a,  6,  and  R'  would  form 
exactly  the  same  triangle  as  a,  b,  and  R,  but  FlG  5a 

for  the  fact  that  the  arrow  head  on  R'  points 

in  the  opposite  direction  to  that  on  R.     Thus  vectors  forming    I  f 
a  closed  triangle  have  a  resultant  equal  to  zero. 

Again,  suppose  that  the  man  while  walking  toward  the  right 
side  of  the  ship,  case  C  (Fig.  5),  and  therefore  having  an  actual 
velocity  R2,  should  throw  a  ball  with  an  equal  velocity  R'z  in  a 
direction  exactly  opposite  to  that  of  R2  (i.e.,  jR'2=— #2).  It 
will  be  evident  at  once  that  the  ball  under  these  circumstances 
would  simply  stand  still  in  the  air  as  far  as  horizontal  motion  is 
concerned.  It  will  be  seen  that  there  are  four  horizontal 
velocities  imparted  to  the  ball.  First,  the  "steam"  velocity 
ai  (Fig.  5a),  second,  the  "drift"  velocity  61,  third,  the  "walking" 
velocity  Ci,  and  fourth,  the  "throwing"  velocity  R'2.  These  four 
velocities,  however,  form  a  closed  polygon  and  the  actual  velocity 
of  the  ball  is  zero.  Hence  we  may  now  make  the  general  state- 
ment that  when  any  number  of  velocities  (or  forces  or  any  other 
vectors)  form  a  Closed  Triangle  or  a  Closed  Polygon,  their  resultant 
is  zero.  This  fact  is  of  great  importance  and  convenience  in  the 
treatment  of  forces  in  equilibrium  and  will  be  made  use  of  in 
some  of  the  problems  at  the  close  of  this  chapter. 

17.  Vectors  in  Equilibrium. — The  method  of  the  preceding 
sections  applies  equally  well  if  the  vectors  involved  are  any  other 


16 


MECHANICS  AND  HEAT 


quantities;  e.g.,  forces,  instead  of  velocities;  and  the  construc- 
tions are  made  in  the  same  way.  This  method  has  many  impor- 
tant applications  in  connection  with  forces,  among  which  is  the 
calculation  of  the  proper  elevation  of  the  outer  rail  on  a  curve 
(Sec.  62)  in  order  that  the  weight,  or  better,  the  thrust  of  a  train 
shall  be  equal  upon  both  rails;  and  the  calculation  of  the  proper 
strength  for  the  different  parts  of  bridges  and  other  structures. 
In  Sec.  16  it  was  shown  that  to  find  in  what  direction  and  with 
what  speed  the  man  must  throw  the  ball  in  order  to  make  its 
actual  velocity  zero,  a  line  R'z  must  be  drawn  equal  to  R2, 
but  oppositely  directed.  R2  is  the  resultant  of  the  three  veloci- 


; 

d 


FIG.  6. 

ties  ai,  bi,  and  Ci,  while  R'z  is  the  Antiresultant  (anti  =  opposed 
to)  or  Equilibrant. 

Thus  it  will  be  seen  that  in  the  graphical  method  the  anti- 
resultant  of  any  number  of  velocities  is  represented  by  a  line 
drawn  from  the  arrow  point  of  the  last  velocity  to  the  beginning 
of  the  first  velocity.  In  other  words,  it  is  represented  by  the 
closing  side  of  the  vector  polygon.  Observe  that  in  this  case  the 
arrow  heads  all  point  in  the  same  way  around  the  polygon; 
while,  if  the  closing  side  is  the  resultant,  its  arrow  head  is  directed 
oppositely  to  all  the  others. 

The  case  of  several  forces  in  equilibrium,  or  so-called  "bal- 
anced forces,"  is  of  special  importance.  The  construction  is 
the  same  as  that  shown  in  Fig.  5a.  Suppose  that  a  body  floating 
in  still  water  is  acted  upon  by  four  horizontal  forces,  whose 


VECTORS  17 

values  are  represented  both  in  magnitude  and  direction  by  the 
lines  a,  b,  c,  and  d  of  A  (Fig.  6).  Let  it  be  required  to  find  the 
magnitude  and  direction  of  a  fifth  force  e',  which  applied  to  the 
body  will  produce  equilibrium,  so  that  the  body  will  have  no 
tendency  to  move  in  any  direction;  in  other  words,  let  us  find 
the  antiresultant  of  a,  b,  c,  and  d.  From  B  (Fig.  6)  we  find 
the  resultant  e,  or  that  single  force  which  would  exactly  replace 
a,  b,  c,  and  d;  i.e.,  which  alone  would  move  the  body  in  the  same 
direction,  and  with  the  same  speed  as  would  these  four  forces. 
The  construction  C  shows  how  e'  is  found.  Obviously,  e' 
and  e  alone  (D,  Fig.  6)  would  produce  equilibrium,  and  since 
e  is  exactly  equivalent  to  a,  b,  c,  and  d,  it  follows  that  a,  b,  c,  d, 
and  e'  produce  equilibrium.  From  E  (Fig.  6)  it  will  be  seen  that 
the  resultant  is  the  same  if  the  vectors  a,  b,  c,  and  d  are  taken  in 
a  different  order. 

This  is  true  for  the  reason  that  wherever,  in  the  construction 
of  the  polygon,  we  choose  to  draw  d,  say,  the  pencil  point  will 
thereby  be  moved  a  definite  distance  to  the  left.  Likewise 
drawing  6  moves  the  pencil  a  definite  distance  to  the  right  and 
downward.  Consequently  the  final  position  of  the  pencil  after 
drawing  lines  a,  b,  c,  and  d,  which  position  determines  the 
resultant  e,  can  in  no  wise  depend  upon  the  order  of  drawing 
these  lines. 

18.  The  Crane. — The  crane,  in  its  simplest  form,  is  shown  in 
Fig.  7.  B  is  a  rigid  beam,  pivoted  at  its  lower  end  and  fastened 
at  its  upper  end  by  a  cable  C  to  a  post  A.  D  is  the  "block  and 
tackle"  for  raising  the  object  L  whose  weight  is  W.  After  the 
object  is  raised,  the  beam  B  may  be  swung  around  horizontally; 
and  then,  by  means  of  the  block  and  tackle,  the  object  may  be 
lowered  and  deposited  where  it  is  wanted.  By  shortening  the 
cable  C  it  is  possible  to  raise  the  weight  higher,  but  the  "sweep" 
of  the  crane  is  of  course  shortened  thereby. 

The  traveling  crane,  used  in  factories,  is  mounted  on  a  "  car- 
riage" which  may  be  run  back  and  forth  on  a  track  sometimes 
extending  the  entire  length  of  the  building,  so  that  a  massive 
machine  weighing  several  tons  may  readily  be  picked  up  and 
carried  to  any  part  of  the  building. 

In  choosing  the  size  of  the  cable  and  the  beam  for  a  crane  as 
sketched,  it  is  necessary  to  know  what  pull  will  be  exerted  on  C, 
and  what  end  thrust  on  B  when  the  maximum  load  is  being 
lifted.  These  two  forces,  c  and  6,  we  shall  now  proceed  to  find. 


18 


MECHANICS  AND  HEAT 


In  Sec.  17  it  was  shown  that  any  number  of  forces  or  any  other 
vectors  in  equilibrium  are  represented  by  a  closed  polygon. 
Three  forces  in  equilibrium  will  accordingly  form  a  closed  tri- 
angle. The  point  0,  at  the  upper  end  of  the  beam  B,  is  obviously 
in  equilibrium  and  is  acted  upon  by  the  three  forces  W,  c,  and 
6;  which  forces,  graphically  represented,  must  therefore  form  a 
closed  triangle.  The  directions  of  6  and  c  are  known  but  not 
their  magnitudes.  W,  however,  is  fully  specified  both  as  to 
direction  and  magnitude.  Hence  the  forces  acting  upon  0 


FIG.  7. 


may  be  represented  as  in  E  (Fig.  7),  or  as  in  F,  since  a  thrust  6 
will  have  the  same  effect  as  an  equal  pull  6.  If  L  weighs  1  ton, 
or  2000  Ibs.,  its  weight  W,  using  as  a  scale  2000  Ibs.  to  the  cm., 
will  be  represented  by  a  line  1  cm.  in  length  (G,  Fig.  7). 
From  the  lower  end  of  W  draw  a  line  b  parallel  to  the  beam,  and 
through  the  other  end  of  W  draw  a  line  c  parallel  to  the  cable. 
The  intersection  of  these  two  lines  at  X  determines  the  magni- 
tude of  both  &  and  c.  For  the  three  forces  have  the  required 
directions,  and  they  also  form  a  closed  triangle,  thus  represent- 
ing equilibrium.  The  length  of  6  in  centimeters  times  2000  Ibs. 


VECTORS 


19 


gives  the  thrust  on  the  beam.     The  value  of  c  is  found  in  the 
same  way.     The  construction  may  also  be  made  as  shown  in  H. 

The  problem  will  be  seen  to  be  simply  this:  Given  one  side  W 
of  a  triangle,  both  in  direction  and  length,  and  the  directions  only 
of  the  other  two  sides  b  and  c;  let  it  be  required  to  construct  the 
triangle. 

19.  Resolution  of  Vectors  into  Components. — The  resolution 
of  a  vector  V  into  two  components,  consists  in  finding  the  magni- 
tude of  two  vectors,  Vi  and  Vz,  whose  directions  are  given,  and  , 
whose  vector  sum  shall  be  the  given  vector  V.  It  is  thus  seen  to  be 
the  converse  of  vector  addition.  The  method  will  be  best  under- 
stood from  one  or  two  applications.  We  shall  here  apply  it 
to  velocities  and  forces,  but  it  applies  equally  well  to  any  other 
vector  quantity. 


FIG.  8. 


FIG.  9. 


A  ship  is  traveling  with  a  uniform  velocity  of  20  mi.  per  hr.  in  a 
direction  somewhat  south  of  east.  An  hour  later  the  ship  is 
18  mi.  farther  east  and  8.7  mi.  farther  south  than  when  first 
observed.  Under  such  circumstances  the  velocity  of  the  ship 
may  be  resolved  into  an  eastward  component  of  18  mi.  per  hr. 
and  a  southward  component  of  8.7  mi.  per  hr.  Had  the  ship 
been  headed  nearly  south,  the  southward  component  would 
have  been  the  larger.  We  shall  next  resolve  a  force  into  two 
components. 

Consider  a  car  B  (Fig.  8)  of  weight  W,  held  by  a  cable  C 
from  running  down  the  inclined  track  A.  Let  it  be  required  to 
find  the  pull  c  that  the  car  exerts  upon  the  cable,  and  also  the 
force  b  that  it  exerts  against  the  track.  The  latter  is  of  course 
at  right  angles  to  the  track,  but  it  is  not  equal  to  the  weight  of 
the  car,  as  might  at  first  be  supposed.  In  fact,  the  weight  of  the 
car  W,  or  the  force  with  which  the  earth  pulls  upon  it,  gives 
rise  to  the  two  forces,  b  and  c.  The  directions  of  6  and  c  are 


Lxiv 


20 


MECHANICS  AND  HEAT 


known,  but  not  their  magnitudes.  In  order  to  find  their  mag- 
nitudes, first  draw  W  to  a  suitable  scale.  Then,  from  the  arrow 
point  of  W,  draw  two  lines,  one  parallel  to  b  and  intersecting  c, 
the  other  parallel  to  c  and  intersecting  b.  These  intersections 
determine  the  magnitudes  of  both  6  and  c,  as  shown.  We  may 
also  determine  b  and  c  by  the  method  used  in  the  solution  of  the 
crane  problem. 

If  the  cable  is  attached  to  a  higher  point,  the  construction  is  as 
shown  in  Fig.  9.  It  will  be  noticed  that  under  these  conditions 
the  c  component  has  become  larger,  and  the  6  component  smaller, 
than  in  Fig.  8.  If  the  cable  is  fastened  directly  above  the  car, 
the  6  component  is  zero;  that  is,  the  car  is  simply  suspended  by 
the  cable. 

In  case  a  force  is  resolved  into  two  components  at  right  angles 
to  each  other,  their  values  may  be  readily 
found  by  the  analytical  method.  Thus  in 
Fig.  8,  c  =  W  sin  8,  and  6  =  W  cos  6.  Note 
that  0i  =  6. 

20.  Sailing  Against  the  Wind.— Al- 
though sailing  "into  the  wind"  by  "tack- 
ing" has  been  practised  by  sea-faring  peo- 
ple from  time  immemorial,  it  is  still  a  puz- 
zle to  many.  Let  AB  (Fig.  10)  represent  a 
sailing  vessel,  CD  its  sail,  CE  the  direction 
in  which  it  is  headed,  and  W  the  direction 
of  the  wind.  If  the  sail  CD  were  friction- 
less  and  perfectly  flat,  the  reaction  of  the 

air  in  striking  it  would  give  rise  to  a  force  F  strictly  at  right  angles 
to  the  sail.  A  push  (force)  against  a  frictionless  surface,  whether 
exerted  by  the  wind  or  by  any  other  means,  must  be  normal  to 
the  surface;  otherwise  it  would  have  a  component  parallel  to 
the  surface,  which  is  impossible  if  there  is  no  friction.  This 
force  F  may  be  resolved  into  the  two  components  'F\  and  F2  as 
shown.  Although  as  sketched,  F2  is  greater  than  the  useful 
component  FI,  nevertheless  the  sidewise  drift  of  the  ship  is  small 
compared  with  its  forward  motion,  because  of  its  greater  resist- 
ance to  motion  in  that  direction.  Making  slight  allowance  for 
this  leeward  drift,  we  have  CE'  for  the  course  of  the  ship. 
Obviously,  in  going  from  C  to  £",  the  ship  goes  the  distance 
CH  "into"  the  wind. 

In  case  the  boat  is  moving  north  at  a  high  velocity,  the  wind,  to 


FIG.  10. 


VECTORS  21 

a  person  on  the  boat,  will  appear  to  come  from  a  point  much  more 
nearly  north  than  it  would  to  a  stationary  observer.  In  other 
words,  the  angle  between  the  plane  of  the  sail  and  the  real  direc- 
tion of  the  wind,  is  always  greater  than  the  angle  between  this 
plane  and  the  apparent  direction  of  the  wind  as  observed  by  an 
occupant  of  the  boat.  It  is,  however,  the  apparent  direction  or, 
perhaps  better,  the  relative  velocity  of  the  wind,  that  determines 
the  reacting  thrust  upon  the  sail.  Hence  strictly,  W  (Fig.  10) 
should  represent  the  apparent  direction  of  the  wind.  It  is  a 
matter  of  common  observation  that,  to  a  man  driving  rapidly 
north,  an  east  wind  appears  to  come  from  a  point 
considerably  north  of  east. 

Because  of  the  very  slight  friction  of  the  wind 
on  the  sail,  F'  is  more  nearly  the  direction  of  the 
push  on  the  sail.  The  useful  component  of  F', 
which  drives  the  ship,  is  obviously  slightly  less 
than  FI  as  found  above  for  the  theoretical  case  of 
no  friction. 

21.  Sailing  Faster  Than  the  Wind. — It  is  pos- 
sible, strange  though  it  may  seem,  to  make  an 
iceboat  travel  faster  than  the  wind  that  drives  pIG 
it.  Let  AB  (Fig.  11)  represent  the  sail  (only) 
of  an  iceboat  which  is  traveling  due  north,  and  v  the  velocity  of 
the  wind.  If  the  runner  friction  were  zero,  so  that  no  power  would 
be  derived  from  the  moving  air,  the  air  would  move  on  unchanged 
in  both  direction  and  speed.  Considering  the  air  that  strikes  at  A, 
this  would  evidently  require  the  sail  to  travel  the  distance  A  A ' 
while  the  wind  traveled  from  A  to  B'.  Hence  the  velocity  v'  of 
the  boat  would  be  AA'/AB'  times  that  of  the  wind,  or  v'/v  = 
AA'/AB'.  The  slight  friction  between  the  runners  and  the  ice 
reduces  this  ratio  somewhat;  nevertheless,  under  favorable  cir- 
cumstances,  an  iceboat  may  travel  twice  as  fast  as  the  wind. 
Velocities  as  high  as  85  mi.  per  hr.  have  been  maintained  for 
short  distances. 

PROBLEMS 

1.  A  balloon  is  traveling  at  the  rate  of  20  miles  an  hour  due  southeast. 
Find  its  eastward  and  southward  components  of  velocity  by  both  the  graph- 
ical and  analytical  methods. 

2.  Find  the  force  required  to  draw  a  wagon,  which  with  its  load  weighs 
2500  Ibs.,  up  a  grade  rising  40  ft.  in  a  distance  of  200  ft.  measured  on  the  grade. 
Neglect  friction,  and  use  the  graphical  method. 


22  MECHANICS  AND  HEAT 

4 

3.  Find  Rlt  case  B  (Fig.  5)  if  0=60°.      (Cos  60°  =0.5).     Use  the  ana- 
lytical method. 

4.  A  boat  which  travels  at  the  rate  of  10  mi.  an  hr.  in  still  water,  is  headed 
S.W.  across  a  stream  flowing  south  at  the  rate  of  4  mi.  an  hr.     A  man  on  the 
deck  runs  at  the  rate  of  7  mi.  an  hr.  toward  a  point  on  the  boat  which  is  due 
east  of  him.     Find  the  actual  velocity  of  the  man  with  respect  to  the  earth, 
and  also  that  of  the  boat.     Use  graphical  method. 

6.  By  the  graphical  method,  find  the  resultant  and  antiresultant  of  the 
following  four  forces:  10  Ibs.  N.,  12  Ibs.  N.E.,  15  Ibs.  E.,  and  8  Ibs.  S. 

6.  If  the  beam  B  (Fig.  7)  is  30  ft.  in  length  and  makes  an  angle  of  45°  with 
the  horizontal,  and  the  guy  cable  C  is  fastened  15  ft.  above  the  lower  end  of 
B,  what  will  be  the  thrust  on  B  and  the  pull  on  C  if  the  load  L  weighs  3000 
Ibs.  ?     Use  the  graphical  method. 

7.  After  a  man  has  traveled  4  miles  east,  and  4  miles  N.,  how  far  must  he 
travel  N.W.  before  he  will  be  due  north  of  the  starting  place,  and  how  far 
will  he  then  be  from  the  starting  place?     Solve  by  both  the  graphical  method 
and  the  analytical  method. 

8.  A  certain  gun,  with  a  light  charge  of  powder,  gives  its  projectile  an 
initial  (muzzle)  velocity  of  300  ft.  per  sec.  when  stationary.     If  this  gun  is 
on  a  car  whose  velocity  is  100  ft.  per  sec.  north,  what  will  be  the  muzzle 
velocity  of  the  projectile  if  the  gun  is  fired  N.?     If  fired  S.?     If  fired  E.? 

9.  A  south  wind  is  blowing  at  the  rate  of  30  mi.  per  hr.     Find,  by  the 
graphical  method  and  also  by  the  analytical  method,  the  apparent  velocity 
of  the  wind  as  observed  by  a  man  standing  on  a  car  which  is  traveling  east 
at  the  rate  of  40  mi.  per  hr. 

10.  The  instruments  on  a  ship  going  due  north  at  the  rate  of  20  miles  an 
hour  record  a  wind  velocity  of  25  miles  per  hour  from  the  N.E.     What  is  the 
actual  velocity  of  the  wind?     Use  the  graphical  method. 

11.  A  tight  rope,  tied  to  two  posts  A  and  B  which  are  20  ft.  apart,  is  pulled 
sidewise  at  its  middle  point  a  distance  of  1  ft.  by  a  force  of  100  Ibs.     By  two 
graphical  methods  (Sec.  18  and  19)  find  the  pull  exerted  on  the  posts.     Solve 
also  by  the  analytical  method. 

12.  Neglecting  friction,  find  the  pull  on  the  cable  and  the  thrust  on  the 
track  in  drawing  a  1000-lb.  car  up  a  45°  incline.    The  cable  is  parallel  to  the 
track. 

13.  Find  the  pull  and  the  thrust  (Prob.  12)  if  the  cable  is  (a)  horizontal; 
(6)  inclined  30°  above  the  horizontal. 


' 


• 


1 

' 

TRANSLATORY  MOTION 


CHAPTER  III 


22.  Kinds  of  Motion. — All  motion  may  be  classed  as  either 
translatory  motion  or  rotary  motion,  or  as  a  combination  of  these 
two.  A  body  has  motion  of  translation  only,  when  any  line 
(which  means  every  line)  in  the  body  remains  parallel  to  its 
original  position  throughout  the  motion.  It  may  also  be  defined 
as  a  motion  in  which  each  particle  of  the  body  describes  a  path 
of  the  same  form  and  length  as  that  of  every  other  particle,  and 
at  the  same  speed  at  any  given  instant;  so  that  the  motion  of 
any  one  particle  represents  completely  the  motion  of  the  entire 
body.  Thus  if  A,  B,  C,  and  D  represent  the  positions  of  a 


FIG.  12. 


FIG.  13. 


triangular  body  (abc)  at  successive  seconds,  it  will  be  noted  that 
in  case  a  moves  a  greater  distance  in  the  second  second 
than  it  does  in  the  first  that  6  and  c  and  all  other  particles  do 
also. 

In  pure  rotary  motion  there  is  a  series  of  particles,  e.g.,  those 
in  the  line  AB  (Fig.  13)  which  do  not  move.  This  line  is  called 
the  axis  of  rotation  of  the  body.  All  other  particles  move  in 
circular  paths  about  this  axis  as  a  center,  those  particles  farthest 
from  the  axis  having  the  highest  velocity. 

Having  obtained  a  clear  notion  of  rotary  motion,  we  may  con- 
sider a  body  to  have  pure  translatory  motion  if  it  moves  from 
one  point  to  another  by  any  path,  however  straight  or  crooked, 
without  any  motion  of  rotation.  The  rifle  ball  has  what  is 

23 


24  MECHANICS  AND  HEAT 

termed  Screw  motion.  The  motion  of  a  steamship  might  seem  to 
be  pure  translatory  motion,  and  indeed  it  closely  approximates 
such  motion  when  the  sea  is  calm.  In  a  rough  sea  its  motion  is 
very  complicated,  consisting  of  a  combination  of  translatory 
motion,  with  to-and-fro  rotation  about  three  axes:  In  the  "roll- 
ing" of  a  ship,  the  axis  is  lengthwise  of  the  ship  or  longitudinal. 
The  "pitching"  of  a  ship  is  a  to-and-fro  rotation  about  a  trans- 
verse axis.  As  the  ship  swerves  slightly  from  its  course,  it 
rotates  about  a  vertical  axis. 

Both  translatory  and  rotary  motion  may  be  either  uniform, 
or  accelerated;  that  is,  the  velocity  may  be  either  constant  or 
changing.  Accelerated  motion  is  of  two  kinds,  uniformly  accel- 
erated and  nonuniformly  accelerated.  Thus  there  are  three  types 
each  of  both  translatory  and  rotary  motion.  Before  discussing 
these  types  of  motion,  it  will  be  necessary  to  define  and  discuss 
velocity  and  acceleration. 

23.  Speed,  Average  Speed,  Velocity,  and  Average  Velocity. — 
As  already  mentioned  (Sec.  13),  speed  is  a  scalar  quantity  and 
velocity  is  a  vector  quantity.  Both  designate  rate  of  motion; 
but  the  former  does  not  take  into  account  the  direction  of  the 
motion,  whereas  the  latter  does. 

Average  speed,  which  may  be  designated  by  s  (read  "  barred  s  ") 
is  given  by 

D 
«  "J  (3) 

in  which  D  is  thejtotal  distance  traversed  by  a  body  in  a  given 
time  t.  Average  velocity  v  is  given  by  the  equation 

-      ^  ft\ 

v  =  y  (4) 

in  which  d  is  the  distance  from  start  to  finish  measured  in  a 
straight  line,  and  t  is  the  time  required.  Observe  that  d  has, 
in  addition  to  magnitude,  a  definite  direction,  and  is  therefore  a 
vector;  whereas  D  is  simply  the  distance  as  measured  along  the 
path  traversed,  which  may  be  quite  tortuous,  and  is  therefore  a 
scalar.  The  Speed  of  a  body  at  any  given  instant  is  the  distance 
which  the  body  would  travel  in  unit  time  if  it  maintained  that 
particular  rate  of  motion;  while  the  Velocity  of  the  body  at  that 
same  instant  has  the  same  numerical  value  as  the  speed,  and 
is  defined  in  the  same  way  except  that  it  must  also  state  the 


TRANSLATORY  MOTION  25 

direction  of  the  motion.     An  example  will  serve  to  further  illus- 
trate the  significance  of  the  above  four  quantities. 

Suppose  that  a  fox  hunt,  starting  at  a  certain  point,  termi- 
nates 10  hrs.  later  at  a  point  20  miles  farther  north.  Suppose 
further  that  during  this  time  the  dog  travels  100  miles.  Then 
d  (Eq.  4)  is  20  miles  due  north  (a  vector),  D  (Eq.  3)  is  100  miles 
(scalar),  ~v  is  2  miles  an  hour  north  (vector),  and  ¥  is  10  miles  an 
hour  (scalar).  If  the  dog's  speed  s  at  a  given  instant  is  15  miles 
an  hour  (often  written  15  mi./hr.  and  called  15  mi.  per  hr.), 
then  an  hour  later,  if  he  continues  to  run  at  that  same  speed,  he 
will  be  15  miles  from  this  point  as  measured  along  the  trail; 
whereas  if  the  dog's  velocity  at  that  same  instant  is  15  miles  per 
hour  east,  then,  an  hour  later,  if  he  maintains  that  same  velocity, 
he  will  be  at  a  point  15  miles  farther  east. 

If  the  hunter  travels  40  miles,  while  a  friend,  traveling  a 
straight  road,  travels  only  20  miles  in  the  ten  hours,  then  the 
hunter's  average  speed  is  twice  that  of  his  friend  and  only  two- 
fifths  that  of  the  dog;  whereas  the  average  velocity  ID  is  the  same 
for  all  three,  viz.,  2  miles  an  hour.  We  thus  see  that  the  aver- 
age velocity  of  a  body  is  that  velocity  which,  unchanged  in 
either  magnitude  or  direction,  would  cause  the  body  to  move  from 
one  point  to  the  other  in  the  same  time  that  it  actually  does 
require. 

24.  Acceleration. — If  a  body  moves  at  a  uniform  speed  in  a 
straight  line  it  is  said  to  have  uniform  velocity,  and  its  velocity 
is  the  distance  traversed  divided  by  the  time  required.  If  its 
speed  is  not  uniform  its  velocity  changes  (in  magnitude),  and 
the  rate  at  which  its  velocity  changes  is  called  the  acceleration,  a. 
If  the  velocity  of  a  body  is  not  changing  at  a  uniform  rate,  then 
the  change  in  velocity  that  occurs  in  a  given  time,  divided  by 
that  time,  gives  the  average  rate  of  change  of  the  velocity  of 
the  body,  or  its  average  acceleration  for  that  time.  Since  the 
second  is  the  unit  of  time  usually  employed,  we  see  that  the 
average  acceleration  is  the  change  (gain  or  loss)  in  velocity  per 
second.  The  acceleration  of  such  a  body  at  any  particular 
instant  is  numerically  the  change  in  velocity  that  would  occur 
in  1  sec.  if  the  acceleration  were  to  have  that  same  value  for 
the  second;  i.e.,  if  the  velocity  were  to  continue  to  change  at 
that  same  rate  for  the  second. 

If  the  velocity  is  increasing,  the  acceleration  is  positive;  if 
decreasing,  it  is  said  to  be  negative.  Thus  the  motion  of  a  train 


26  MECHANICS  AND  HEAT 

when  approaching  a  station  with  brakes  applied,  is  accelerated 
motion.  As  it  starts  from  the  station  it  also  has  accelerated 
motion,  but  in  this  case  the  acceleration  is  positive,  since  it  is  in 
the  direction  of  the  velocity;  while  in  the  former  case,  the 
acceleration  is  negative. 

If  the  acceleration  of  a  body  is  constant,  for  example  if  the  body 
continues  to  move  faster  and  faster,  and  the  increase  in  velocity 
each  succeeding  second  or  other  unit  of  time  is  the  same,  its 
motion  is  said  to  be  uniformly  accelerated.  Thus  if  the  velocity 
of  a  body  expressed  in  feet  per  second,  e.g.,  the  velocity  of  a 
street  car,  has  the  values  10,  12,  14,  16,  18,  etc.,  for  successive 
seconds;  then  the  acceleration  a  for  this  interval  is  constant, 
and  has  the  value  2  ft.  per  sec.  per  sec.,  or 

a  =  2  ft.  per  sec.  per  sec.  (also  written  2  —  ^,) 

sec.2 

If  a  certain  train  is  observed  to  have  the  above  velocities  for 
successive  minutes,  then  the  motion  of  the  train  is  uniformly 
accelerated,  since  its  acceleration  is  constant;  but  it  is  less  than 
above  given  for  the  street  car,  in  fact,  1/60  as  great,  or  2  ft.  per 
sec.  per  min. ;  that  is, 

2  ft 

a  =  2  ft.  per  sec.  per  min.  (also  written  ,    ) 

sec.  mm. 

This  means  that  the  gain  of  velocity  each  minute  is  2  ft.  per  sec. 
A  freely  falling  body,  or  a  car  running  down  a  grade  due  to  its 
weight  only,  are  examples  of  uniformly  accelerated  motion.  In 
order  that  a  body  may  have  accelerated  motion,  it  must  be  acted 
upon  by  an  applied  or  external  force  differing  from  that  required 
\  to  overcome  all  friction  effects  upon  the  body. 

25.  Accelerating  Force. — Force  may  be  defined  as  that  which 
produces  or  tends  to  produce  change  in  the  velocity  of  a  body,  to 
which  it  is  applied;  i.e.,  force  tends  to  accelerate  a  body.  A  force 
may  be  applied  to  a  body  either  as  a  push  or  a  pull.  It  has  been 
shown  experimentally  that  it  requires,  for  example,  exactly  twice 
as  great  a  force  to  give  twice  as  great  an  acceleration  to  a  given 
mass  which  is  perfectly  free  to  move;  and  also  that  if  the  mass  be 
doubled  it  requires  twice  as  much  force  to  produce  the  same  ac- 
celeration. In  other  words,  the  force  (F)  is  proportional  to  the 
resulting  acceleration  (a),  and  also  proportional  to  the  mass 


TRANSLATORY  MOTION  27 

(M)  of  the  body  accelerated.  These  facts  are  expressed  by  the 
equation 

F  =  Ma  (5) 

For,  to  increase  a  n-fold,  F  must  be  increased  n-fold;  in  other 
words,  the  resulting  acceleration  of  a  body  is  directly  propor- 
tional to  the  applied  force,  and  is  also  inversely  proportional  to 
the  mass  of  the  body. 

Eq.  5  is  sometimes  written  F  =  kMa.  If  the  units  of  force, 
mass,  and  acceleration  are  properly  chosen  (see  below),  k  becomes 
unity  and  may  be  omitted. 

Units  of  Force. — Imagine  the  masses  now  to  be  considered,  to 
be  perfectly  free  to  move  on  a  level  frictionless  surface,  and  let 
the  accelerating  force  be  horizontal.  Then  the  unit  force  in  the 
metric  system,  the  Dyne,  is  that  force  which  will  give  unit  mass 
(1  gm.)  unit  acceleration  (1  cm.  per  sec.  per  sec.);  while  in  the 
British  system,  unit  force,  the  Poundal,  is  that  force  which  will 
give  unit  mass  (1  Ib.) -unit  acceleration  (1  ft.  per  sec.  per  sec.). 
Thus,  to  cause  the  velocity  of  a  10-gm.  mass  to  change  by  4  cm. 
per  sec.  in  1  sec.;  i.e.,  to  give  it  an  acceleration  of  4  cm.  per  sec. 
per  sec.,  will  require  an  accelerating  force  of  40  dynes,  as  may  be 
seen  by  substituting  in  Eq.  5. 

The  relation  between  these  units  and  the  common  gravita- 
tional units,  the  gram  weight  and  the  pound  weight,  will  be  ex- 
plained under  the  study  of  gravitation  (Sec.  32) ;  but  we  may  here 
simply  state  without  explanation  that  1  gram  weight  is  equal  to 
980  dynes  (approx.),  and  that  1  pound  weight  is  equal  to  32.2 
poundals  (approx.). 

In  general,  only  a  part  of  the  force  applied  to  a  body  is  used 
in  accelerating  it,  the  remainder  being  used  to  overcome  friction 
or  other  resistance.  The  part  that  is  used  in  producing  accelera- 
tion is  called  the  Accelerating  Force.  It  should  be  emphasized 
that  Eq.  5  holds  only  if  F  is  the  accelerating  force.  Thus  if  a 
stands  for  the  acceleration  in  the  motion  of  a  train,  and  M  for 
the  mass  of  the  train,  then  F  is  not  the  total  pull  exerted  by  the 
drawbar  of  the  engine,  but  only  the  excess  pull  above  that  needed 
to  overcome  the  friction  of  the  car  wheels  on  axle  bearings  and 
on  the  track,  air  friction,  etc.  If  an  8000-lb.  pull  is  just  sufficient 
to  maintain  the  speed  of  a  certain  train  at  40  miles  an  hour  on  a 
level  track,  then  a  pull  of  9000  Ibs.  would  cause  its  speed  to  in- 
crease, and  7000  Ibs.,  to  decrease.  The  accelerating  force,  i.e., 


28  MECHANICS  AND  HEAT 

the  F  of  Eq.  5,  would  be  1000  Ibs.,  i.e.,  32,200  poundals,  in  each 
case. 

In  the  case  of  a  freely  falling  body,  the  accelerating  force  is 
of  course  the  pull  of  the  earth  upon  the  body,  or  its  weight;  while 
in  the  case  of  a  lone  car  running  down  a  grade,  it  is  the  component 
of  the  car's  weight  parallel  to  the  grade  (see  Fig.  8),  minus  the 
force  required  to  overcome  friction,  that  gives  the  accelerating 
force.  We  may  now  make  the  statement  that  when  a  body  is  in 
motion  its  velocity  will  not  change  if  the  force  applied  is  just  suffi- 
cient to  overcome  friction;  while  if  the  force  is  increased,  the 
velocity  will  increase,  and  the  acceleration  will  be  positive  and 
proportional  to  this  increase  or  excess  of  force.  If  the  applied 
force  is  decreased  so  as  to  become  less  than  that  needed  to 
overcome  friction,  then,  of  course,  the  velocity  decreases,  and  the 
acceleration  is  negative  and  proportional  to  the  deficiency  of 
the  applied  force. 

•"  26.  Uniform  Motion  and  Uniformly  Accelerated  Motion. — 
This  subject  will  be  best  understood  if  discussed  in  connection 
with  a  specific  example.  Suppose  that  a  train,  traveling  on  a 
straight  track  and  at  a  uniform  speed  from  a  town  A  to  a  town  B 
20  miles  north  of  A,  requires  30  minutes  time.  In  this  case  its 
velocity 

_  distance  traversed  _  d  _20  miles 
time  required      =  t  =  30  min. 

or  2/3  of  a  mile  per  min.  north.  Since  the  velocity  is  constant, 
the  train  is  said  to  have  Uniform  Motion.  If  the  track  is  level, 
the  pull  on  the  drawbar  of  the  engine  must  be  just  sufficient  to 
overcome  friction,  since  there  is  no  acceleration  and  hence  no 
accelerating  force.  Thus,  uniform  motion  may  be  defined  as 
the  motion  of  a  body  which  experiences  no  acceleration.  This 
train  would  have  to  be  a  through  train;  for  if  it  is  a  train  that 
stops  at  A,  its  velocity  just  as  it  leaves  A  would  be  increasing; 
i.e.,  there  would  be  an  acceleration.  Consequently  there  would 
have  to  be  an  accelerating  force;  that  is,  the  pull  on  the  drawbar 
would  have  to  be  greater  than  the  force  required  to  overcome 
friction.  In  this  case  the  motion  would  be  accelerated  motion. 

In  case  the  accelerating  force  is  constant,  for  example,  if  the 
pull  on  the  drawbar  exceeds  the  force  required  to  overcome 
friction  by,  say  4000  Ibs.  constantly  for  the  first  minute,  then  the 
acceleration  (a)  is  constant  or  uniform,  and  the  motion  for  this 


TRANSLATOR?  MOTION  29 

first  minute  would  be  Uniformly  Accelerated  Motion.  For,  from 
F  =  Ma  (Eq.  5),  we  see  that  if  the  accelerating  force  F  (here 
4000  Ibs.)  is  constant,  a  will  also  be  constant;  i.e.,  the  velocity  of 
the  train  will  increase  at  a  uniform  rate.  As  a  rule,  this  excess 
pull  is  not  constant,  so  that  the  acceleration  varies,  and  the  train 
has  nonuniformly  accelerated  motion. 

Let  us  further  consider  the  motion  of  the  above  train  if  the 
accelerating  force  is  constant,  and  its  motion,  consequently, 
uniformly  accelerated.  Suppose  that  its  velocity  as  it  passes  a 
certain  bridge  is  20  ft.  per  sec.  and  that  we  represent  it  byy0; 
while  its  velocity  10  seconds  later  (or  t  sec.  later)  is  34.6  ft.  per 
sec.,  represented  by  vt.  Its  total  change  of  velocity  in  this  time 
t  is  vt  —  vu,  hence  the  acceleration 

vt-Vo       34.6-20 
a  =  — -. —  =  — 10 =  1.46  ft.  per  sec.  per  sec.  (6) 

It  is  customary  to  represent  the  velocity  first  considered  by  va, 
and  the  velocity  t  seconds  later  by  vt,  as  we  have  here  done.  If 
we  first  consider  the  motion  of  the  train  just  as  it  starts  from  A, 
i.e.,  as  it  starts  from  rest,  then  v0  is  zero,  and  vt  is  its  velocity  t 
seconds  after  leaving  A.  If  t  is  60  sec.,  then  vt  is  the  velocity  of 
the  train  60  seconds  after  leaving  A. 

Let  us  suppose  that  one  minute  after  leaving  A  (from  rest) 
the  velocity  of  the  train  is  60  miles  per  hour.  This  is  the  same 
as  1  mile  per  min.  or  88  ft.  per  sec.  The  total  change  in  velocity 
in  the  first  minute  is  then  60  miles  per  hour,  and  hence  the  accel- 
eration is  60  miles  per  hour  per  minute,  or 

a  =  60  miles  per  hr.  per  min. 
This  same  acceleration  is  1  mile  per  minute  per  minute  or 

a  =  l  mi.  per  min.  per  min. 
It  is  also  88  ft.  per  second  per  minute,  or 

QO 

a  =  88  ft.  per  sec.  per  min.  =77^  ft.  per  sec.  per  sec. 

OU 

=  1.46  ft.  per  sec.  per  sec. 

This  equation  states  that  the  change  of  velocity  in  one  minute 
is  88  ft.  per  sec.,  while  in  one  second  it  is  of  course  1/60  of 
this,  or  1.46  ft.  per  sec.  Ten  seconds  after  the  train  leaves  A, 


30  MECHANICS  AND  HEAT 

its  velocity  is  10X1.46  or  14.6  ft.  per  sec.     Observe  that  when 
v0  is  zero,  Eq.  6  may  be  written 

vt  =  at  (7) 

27.  Universal  Gravitation. — Any  two  masses  of  matter  exert 
upon  each  other  a  force  of  attraction.  This  property  of  matter  is 
called  Universal  Gravitation.  Thus  a  book  held  in  the  hand 
experiences  a  very  feeble  upward  pull  due  to  the  ceiling  and  other 
material  above  it;  side  pulls  in  every  direction  due  to  the  walls, 
etc.;  and  finally,  a  very  strong  downward  pull  due  to  the  earth. 
This  downward  pull  or  force  is  the  only  one  that  is  large  enough 
to  be  measured  by  any  ordinary  device,  and  is  what  is  known  as 
the  weight  of  the  body. 

That  there  is  a  gravitational  force  of  attraction  exerted  by 
every  body  upon  every  other  body,  was  shown  experimentally 
by  Lord  Cavendish.  A  light  rod  with  a  small  metal  ball  at 
each  end  was  suspended  in  a  horizontal  position  by  a  vertical 
wire  attached  to  its  center.  A  large  mass,  say  A,  placed  near 
one  of  these  balls  B  and  upon  the  same  level  with  it,  was  found 
to  exert  upon  the  ball  a  slight  pull  which  caused  the  rod  to  rotate 
and  twist  the  suspending  wire  very  slightly.  Comparing  this 
slight  pull  on  B  due  to  A,  with  the  pull  of  the  earth  upon  5,  i.e., 
with  B's  weight,  Cavendish  was  able  to  compute  the  mass  of  the 
earth.  In  popular  language,  he  Weighed  the  Earth. 

From  the  mass  of  the  earth  and  its  volume  Lord  Cavendish 
determined  the  average  density  of  the  earth  to  be  about  5.5 
times  that  of  water.  The  surface  soil  and  surface  rocks — 
sandstone,  limestone,  etc. — have  an  average  density  of  but  2.5 
times  that  of  water.  Hence  the  deeper  strata  of  the  earth  are 
the  more  dense,  and  consequently  as  a  body  is  lowered  into  a 
mine  and  approaches  closer  and  closer  to  the  more  dense  mate- 
rial, its  weight  might  be  expected  to  increase.  The  upward 
attraction  upon  the  body  exerted  by  the  overlying  mass  of  earth 
and  rocks  should  cause  its  weight  to  decrease.  The  former  more 
than  offsets  the  latter,  so  that  there  is  a  slight  increase  in  the 
weight  of  a  body  as  it  is  carried  down  into  a  deep  mine. 

Newton's  Law  of  Gravitation. — Sir  Isaac  Newton  was  the  first  to  express 
clearly  the  law  of  universal  gravitation  by  means  of  an  equation.  He 
made  the  very  logical  assumption  that  the  attractive  gravitational 
force  (F)  exerted  between  two  masses  MI  and  M 2,  when  placed  a  distance 


<r  W*    ^ 

-  frKtt 


TRANSLATORY  MOTION  31 

d  apart,  would  be  proportional  to  the  product  of  the  masses,  and 
inversely  proportional  to  the  square  of  the  distance  between  them  (Sec. 
28),  i.e., 

F_kMM* 
d* 

If,  in  this  equation,  MI  and  M2  are  expressed  in  grams,  the  distance  in 
centimeters,  and  F  in  dynes,  then  k,  the  proportionality  constant  or 
proportionality  factor  (Sec.  28)  is  shown  by  experiment  to  be 
0.0000000666.  If  Mi,  Mz,  and  d  are  all  unity,  then  F  =  k.  In  other 
words,  the  gravitational  attraction  between  two  1-gm.  masses  when  1  cm. 
apart  is  0.0000000666  dynes.  Since  the  dyne  is  a  small  force,  this  will 
be  seen  to  be  a  very  small  force.  Lord  Cavendish  used  this  equation  in 
computing  his  results. 

28.  The  Law  of  the  Inverse  Square  of  the  Distance.— This 
law  is  one  of  the  most  important  laws  of  physics  and  has  many 
applications,  a  few  of  which  we  shall  now  consider.  We  are 
all  familiar  with  the  fact  that  as  we  recede  from  a  source  of  light, 
for  example  a  lamp,  the  intensity  of  the  light  decreases.  That 
the  intensity  of  illumination  at  a  point  varies  inversely  as  the 
square  of  the  distance  from  that  point  to  the  light  source,  has 
been  repeatedly  verified  by  experiment,  and  it  may  also  be  dem- 
onstrated by  a  simple  line  of  reasoning  as  follows:  Imagine  a 
lamp  which  radiates  light  equally  in  all  directions,  to  be  placed 
first  at  the  center  of  a  hollow  sphere  of  1  ft.  radius,  and  later  at 
the  center  of  a  similar  hollow  sphere  whose  radius  is  3  ft. 
In  each  case  the  hollow  sphere  would  receive  all  of  the  light 
emitted  by  the  lamp,  but  in  the  second  case  this  light  would  be 
distributed  over  9  times  as  much  surface  as  in  the  first.  Hence, 
the  illumination  would  be  1/9  as  intense,  and  we  have  therefore 
proved  that  the  intensity  of  illumination  varies  inversely  as  the 
square  of  the  distance  from  the  lamp. 

An  exactly  similar  proof  would  show  that  the  same  law  applies 
in  the  case  of  heat  radiation,  or  indeed  in  the  case  of  any  effect 
which  acts  equally  in  all  directions  from  the  source.  This 
law  has  been  shown  to  hold  rigidly  in  the  case  of  the  gravita- 
tional attraction  between  bodies,  for  example  between  the  differ- 
ent members  of  the  solar  system. 

Proportionality  Factor. — In  all  cases  in  which  one  quantity  is  propor- 
tional to  another,  the  fact  may  be  stated  by  an  equation  if  we  introduce 
a  proportionality  factor  (k).  Thus  the  weight  of  a  certain  quantity  of 


32  MECHANICS  AND  HEAT 

water  is  proportional  to  its  volume;  i.e.,  3  times  as  great  volume  will 
have  3  times  as  great  weight,  and  so  on.  We  may  then  write 

W«V,  but  not  W=V 
We  may,  however,  write 

W  =  kV 

in  which  k  is  called  the  proportionality  factor.  In  this  case  k  (in  the 
English  system)  would  be  numerically  the  weight  of  a  cubic  foot  of 
water,  or  62.4  (1  cu.  ft.  weighs  62.4  Ibs.),  V  being  the  number  of  cubic 
feet  whose  weight  is  sought. 

We  may  add  another  illustration  of  the  use  of  the  proportionality 
factor.  We  have  just  seen  that  the  illumination  (/)  at  a  point  varies 
inversely  as  the  square  of  the  distance  from  the  source.  We  also  know 
that  it  should  vary  as  the  candle  power  (C.P.)  of  the  source.  Hence  we 
may  write 

.    C.P.  .CP. 

/---,  or  7  =  *-^- 

A  third  illustration  has  already  been  given  at  the  close  of  Sec.  27. 

29.  Planetary  Motion. — The  earth  revolves  about  the  sun  once 
a  year  in  a  nearly  circular  orbit  of  approximately  93,000,000 
miles  radius.  The  other  seven  planets  of  the  solar  system  have 
similar  orbits.  The  planets  farthest  from  the  sun  have,  of  course, 
correspondingly  longer  orbits,  and  they  also  travel  more  slowly; 
so  that  their  "year"  is  very  much  longer  than  ours.  Thus 
Neptune,  the  most  distant  planet,  requires  about  165  years  to 
traverse  its  orbit,  while  Mercury,  which  is  the  closest  planet  to 
the  sun,  has  an  88-day  "year."  The  moon  revolves  about  the 
earth  once  each  lunar  month  in  an  orbit  of  approximately  240,000 
miles  radius.  Several  of  the  planets  have  moons  revolving 
about  them  while  they  themselves  revolve  about  the  sun. 

If  a  stone  is  whirled  rapidly  around  in  a  circular  path  by  means 
of  an  attached  string,  we  readily  observe  that  a  considerable  pull 
must  be  exerted  by  the  string  to  cause  the  stone  to  follow  its 
constantly  curving  path  (Sec.  58).  In  the  case  of  the  earth  and 
the  other  planets,  it  is  the  gravitational  attraction  between  planet 
and  sun  that  produces  the  required  inward  pull.  Our  moon  is 
likewise  held  to  its  path  by  means  of  the  gravitational  attraction 
between  the  earth  and  the  moon.  The  amount  of  pull  required 
to  keep  the  moon  in  its  course  has  been  computed,  and  found  to 
be  in  close  agreement  with  the  computed  gravitational  pull  that 


TRANSLATORY  MOTION  33 

the  earth  should  exert  upon  a  body  at  that  distance.  In  comput- 
ing the  latter  it  was  assumed  that  the  inverse  square  law  (Sec. 
28)  applied. 

Since  the  moon  is  approximately  60  times  as  far  from  the  center 
of  the  earth  as  we  are,  it  follows  that  the  pull  of  the  earth  upon  a 
pound  mass  at  the  moon  is  (I/GO)2  or  1/3600  pound.  By  means 
of  the  formulas  developed  in  Sec.  58,  the  student  can  easily  show 
that  this  force  would  exactly  suffice  to  cause  the  moon  to  follow 
its  constantly  curving  path  if  it  had  only  one  pound  of  mass.  Since 
the  mass  of  the  moon  is  vastly  greater  than  one  pound,  it  requires 
a  correspondingly  greater  force  or  pull  to  keep  it  to  its  orbit,  but 
its  greater  mass  also  causes  the  gravitational  pull  between  it  and 
the  earth  to  be  correspondingly  greater  so  that  this  pull  just 
suffices. 

30.  The  Tides. — A  complete  discussion  of  the  subject  of  tides 
is  beyond  the  scope  of  this  work,  but  a  brief  discussion  of  this 
important  phenomenon  may  be  of  interest.  Briefly  stated,  the 

a  ^^_ 

/"          Moon   "~^\ 


FIG.  14.  FIG.  15. 

$%£  &< 

main  cause  of  tides  is  the  fact  that  the  gravitational  attraction  of        (>  t 
the  moon  upon  unit  mass  is  greater  for  the  ocean  upon  the  side  of 
the  earth  toward  it,  than  for  the  main  body  of  the  earth;  while  ,VLJ 
for  the  ocean  lying  upon  the  opposite  side  of  the  earth,  it  is  less. 
This  follows  directly  from  a  consideration  of  the  law  of  inverse 
squares  (Sec.  28). 

This  difference  in  lunar  gravitational  attraction  tends  to  heap 
the  water  slightly  upon  the  side  of  the  earth  toward  it  and  also 
upon  the  opposite  side;  consequently  if  the  earth  always  presented 
the  same  side  to  the  moon,  these  two  "heaps"  would  be  perma- 
nent and  stationary  (Fig.  14).  As  the  earth  rotates  from  west 
to  east,  however,  these  two  "heaps"  or  tidal  waves  travel  from 


34  MECHANICS  AND  HEAT 

east  to  west  around  the  earth  once  each  lunar  day  (about  24  hrs. 
50  min.),  tending,  of  course,  to  keep  directly  under  the  moon. 
Due  to  the  inertia  of  the  water,  the  tidal  wave  lags  behind 
the  moon;  so  that  high  tide  does  not  occur  when  the  moon  is 
overhead  (Fig.  14),  but  more  nearly  at  the  time  it  is  setting,  and 
also  when  it  is  rising  (Fig.  15).  Since  the  moon  revolves  about 
the  earth  from  west  to  east  in  approximately  28  days,  we  see 
why  the  lunar  day,  moonrise  to  moonrise,  or  strictly  speaking, 
"moon  noon"  to  "moon  noon,"  is  slightly  longer  than  the  solar 
day  (Sec.  4). 

Every  body  of  the  solar  system,  so  far  as  known,  except  Nep- 
tune's moon,  revolves  in  a  counterclockwise  direction  both  about 
its  axis  and  also  in  its  orbit  as  viewed  from  the  North  Star.  Hence 
the  arrows  a,  b,  c,  and  d  respectively  represent  the  motion  of  the 
moon,  rotation  of  the  earth,  motion  of  the  tides,  and  apparent 
motion  of  the  moon  with  respect  to  the  earth.  Consequently, 
according  to  this  convention,  the  moon  rises  at  the  left  and  sets 
at  the  right,  which  is  at  variance  with  the  usual  geographical 
convention. 

Although  the  sun  has  a  vastly  greater  mass  than  the  moon,  its 
much  greater  distance  from  the  earth  reduces  its  tidal  effect  to 
less  than  half  that  of  the  moon.  During  new  moon,  when  the 
sun  and  moon  are  on  the  same  side  of  the  earth,  or  at  full  moon, 
when  on  opposite  sides,  their  tidal  effects  are  evidently  additive,  and 
therefore  produce  the  maximum  high  tides  known  as  Spring 
tides.  During  first  quarter  and  last  quarter  their  tidal  effects  are 
subtractive,  giving  the  minimum  high  tides  or  Neap  tides.  For 
if  the  sun  were  in  the  direction  S  (Fig.  14)  it  would  tend  to  pro- 
duce high  tide  at  e  and/,  and  low  tide  at  c  and  h. 

On  small  islands  in  mid-ocean,  the  tidal  rise  is  but  a  few  feet; 
while  in  funnel-shaped  bays  facing  eastward,  such  as  the  Bay  of 
Fundy,  for  example,  it  is  from  40  to  50  feet. 

If  the  earth  were  completely  surrounded  by  an  ocean  of  uniform  depth, 
the  above  simple  theory  would  explain  the  behavior  of  the  tides.  Under 
such  circumstances  tides  would  always  travel  westward.  The  irregular 
form  and  varying  depth  of  the  ocean  make  the  problem  vastly  more 
complex.  Thus  the  tide  comes  to  the  British  Isles  from  the  south- 
west. (Ency.  Brit.).  This  tide,  which  is  simply  a  large  long  wave  pro- 
duced by  the  true  tidal  effect  in  a  distant  portion  of  the  open  ocean,  first 
reaches  the  west  coasts  of  Ireland  and  England,  and  then,  passing 
through  the  English  Channel,  reaches  London  several  hours  later. 


TRANSLATORY  MOTION  35 

31.  Acceleration  of  Gravity  and  Accelerating  Force  in  Free 
Fall. — Since  the  earth  exerts  the  same  pull  upon  a  body  whether 
at  rest  or  in  motion,  it  will  be  evident  that  the  accelerating  force 
in  the  case  of  a  falling  body  is  simply  its  weight  W,  and  hence  we 
have  from  Eq.  5,  Sec.  25. 

W  =  Ma,  or  W=Mg  (8) 

in  which  M  is  the  mass  of  the  falling  body,  and  g  is  its  accelera- 
tion. It  is  customary  to  use  g  instead  of  a  to  designate  the 
acceleration  of  gravity,  i.e.,  the  acceleration  of  a  freely  falling 
body.  From  Eq.  8  we  see  that  g  =  W/M,  and  since  a  mass  n 
times  as  large  has  n  times  as  great  weight,  g  must  be  constant; 
i.e.,  a  10-lb.  mass  should  fall  no  faster  than  a  1-lb.  mass,  neglect- 
ing air  friction.  If  it  were  not  for  air  friction,  a  feather  would  fall 
just  as  fast  as  a  stone.  This  has  been  demonstrated  by  placing 
a  coin  and  a  feather  in  a  glass  tube  ("guinea  and  feather"  ex- 
periment) and  then  exhausting  the  air  from  the  tube  by  means 
of  an  air  pump.  Upon  inverting  the  tube,  it  is  found  that  the 
coin  and  the  feather  fall  equally  fast;  hence  they  must  both  ex- 
perience the  same  constant  acceleration.  From  Eq.  8  it  follows 
that  g  varies  in  value  with  change  of  altitude  or  latitude  just 
as  does  the  weight  W  of  a  body  (Sec.  11). 

Since  the  acceleration  of  gravity,  g,  represents  the  rate  at 
which  any  falling  body  gains  velocity,  it  is  at  once  evident  that 
it  is  a  very  important  constant.  Its  value  has  been  repeatedly 
determined  with  great  care,  and  it  has  been  found  that 

0  =  980.6  cm.  per  sec.  per  sec.  (9) 

for  points  whose  latitude  is  about  45°.  For  points  farther  north 
it  is  slightly  greater  than  this  (983.2  at  pole);  and  for  points 
farther  south,  slightly  less  (978  at  equator.)  The  above  equa- 
tion states  that  in  one  second  a  falling  body  acquires  an  addi- 
tional velocity  of  980.6  cm.  per  sec.  Since  980.6  cm.  per  sec.  = 
32.17  ft.  per  sec.,  we  have 

gr  =  32.17  ft.  per  sec.  per  sec.  (9a) 

We  may  define  the  Acceleration  of  Gravity  as  the  rate  of  change 
of  velocity  of  a  freely  falling  body;  hence  it  is  numerically  the 
additional  velocity  acquired  by  a  body  in  each  second  of  free 
fall.  If  it  were  not  for  air  friction,  a  body  would  add  this  32.17 
ft.  per  sec.  (980.6  cm.  per  sec.)  to  its  velocity  every  second,  h<?w- 


36  MECHANICS  AND  HEAT 

ever  rapidly  it  might  be  falling.  Though  a  close  study  of  the 
effects  of  air  friction  upon  the  acceleration  is  beyond  the  scope 
of  this  course,  we  readily  see  that  when  a  falling  body  has  ac- 
quired such  a  velocity  that  the  air  friction  resisting  its  fall  is 
equal  to  one-third  of  its  weight,  then  only  two-thirds  of  its  weight 
remains  as  the  accelerating  force.  Its  acceleration  would  then, 
of  course,  be  only  two-thirds  g.  When  a  falling  body,  for  exam- 
ple a  hailstone,  has  acquired  such  a  velocity  that  the  air  friction 
encountered  is  just  equal  to  its  weight,  then  its  entire  weight  is 
used  in  overcoming  friction,  the  accelerating  force  acting  upon 
it  has  become  zero,  and  its  acceleration  is  zero;  i.e.,  it  makes  no 
further  gain  in  velocity. 

32.  Units  of  Weight  and  Units  of  Force  Compared. — From 
Eq.  5  (Sec.  25)  we  see  that  the  logical  unit  of  force  is  that  force 
which  will  give  unit  mass  unit  acceleration,  or  unit  change  of 
velocity  in  unit  time.  Hence,  in  the  metric  system,  unit  force, 
or  the  Dyne  (See  also  "Units  of  Force,"  Sec.  25),  is  that  force 
which  will  give  one  gram  mass  an  acceleration  of  1  cm.  per  sec. 
per  sec.,  i.e.,  a  change  in  velocity  of  1  cm.  per  sec.  in  a 
second.  In  the  case  of  a  gram  mass  falling,  the  accelerating  force 
is  a  gram  weight,  and  the  velocity  imparted  to  it  in  one  second 
is  found  by  experiment  (in  latitude  45°)  to  be  980.6  cm.  per  sec. 
(Sec.  31);  whence  g  equals  980.6  cm.  per  sec.  per  sec.  It 
follows  at  once,  then,  that  a  gram  weight  equals  980.6  dynes, 
since  it  produces  when  applied  to  a  gram  mass  980.6  times  as 
great  an  acceleration  as  the  dyne  does.  Likewise  in  the  British 
system,  unit  force  (the  Poundal)  is  that  force  which  will  give  unit 
mass  (the  pound)  unit  velocity  (1  ft.  per  sec.)  in  unit  time  (the 
second).  But  in  the  case  of  a  pound  mass  freely  falling,  the 
accelerating  force  is  one  pound  weight,  and  this  force,  as  experi- 
ment shows,  imparts  to  it  a  velocity  of  32.17  ft.  per  sec.  in  one 
second.  It  follows  at  once  that  one  pound  force,  or  one  pound 
weight,  equals  32.17  poundals,  since  it  produces  32.17  times 
as  great  acceleration  with  the  same  mass  (see  Eq.  5). 

The  poundal  and  the  dyne  are  the  absolute  units  of  force.  The 
pound,  ton,  gram,  kilogram,  etc.,  are  some  of  the  units  of  force  in 
common  use.  Forces  are  measured  by  spring  balances  and 
other  weighing  devices. 

In  Eq.  8,  the  weight  is  expressed  in  absolute  units;  in  which 
case  W  =  Mg.  If  W  is  expressed  in  grams  weight  or  pounds 
weight,  then  we  have  simply  W  =  M  (numerically),  i.e.,  a  100- 


TRANSLATORY  MOTION  37 

gm.  mass  weighs  100  grams,  or  98,060  dynes.    Likewise  a  10-lb. 
mass  weighs  10  Ibs.,  or  321.7  poundals  (latitude  45°). 

The  Engineer's  Units  of  Force  and  Mass. — In  engineering  work 
the  pound  is  used  as  the  unit  of  force  instead  of  the  poundal. 
Transposing  Eq.  8,  Sec.  31,  we  have  M  =  W/g.  Now  in  physics, 
W  is  expressed  in  poundals,  M  being  in  pounds,  while  in  engi- 
neering work  W  is  expressed  in  pounds.  Since  the  pound  is  32.17 
times  as  large  a  unit  as  the  poundal,  M  must  be  expressed  in 
the  engineering  system  in  a  unit  32.17  times  as  large  as  the 
pound  mass  (close  approximation).  This  32.17-lb.  mass  is 
sometimes  called  the  Slug. 

As  a  summary,  let  us  write  the  equation  F  =  Ma,  and  the  simi- 
lar equation  restricted  to  gravitational  acceleration;  namely, 
W  =  Mg,  indicating  the  units  for  each  symbol  in  all  three  sys- 
tems— the  Metric,  the  British,  and  the  Engineering  systems. 

Metric  System: 

F  =  Ma  and  W  =  Mg,  i.e.,  F'or  W  (dynes) 

=  M (gm.)  X«  or  g  (cm.  per  sec.  per  sec.). 

British  System: 

F  =  Ma  and  W  =  Mg,  i.e.,  F  or  W  (poundals) 

=  M(lbs.)  Xa  or  g  (ft.  per  sec.  per  sec.). 

Engineering  System: 

W 
F  =  —  a=(Ma)  and  W  =  Mg,  i.e.,  F  or  W  (pounds) 

{/ 

=  Af(slugs)  X«  or  g  (ft.  per  sec.  per  sec.). 

Thus,  practically,  the  engineering  system  differs  from  the 
British  system  in  that  the  units  of  mass,  force,  and  weight  are 
32.17  times  as  large  as  the  corresponding  units  in  the  British  system. 

Some  regret  that  the  engineering  system  was  ever  introduced. 
It  is  now  firmly  established,  however,  and  the  labor  involved  in 
mastering  this  third  system  is  very  slight,  indeed,  if  the  British 
system  is  thoroughly  understood.  Furthermore,  this  system 
has  in  some  cases  certain  advantages. 

Observe  that  the  word  "pound"  is  used  for  the  unit  of  mass 
and  also  for  one  of  the  units  of  force.  Having  defined  the  pound 
force  as  the  weight  of  a  pound  mass,  we  may  (and  frequently  do) 
use  it  (the  pound  force)  as  the  unit  in  measuring  forces  which 
have  absolutely  nothing  to  do  with  either  mass  or  weight.  Thus 
in  stretching  a  clothes  line  with  a  force  of,  say  50  Ibs.,  it  is  clear 
that  this  50-lb.  force  has  nothing  to  do  with  the  mass  or  weight 
of  the  clothes  line,  or  post,  or  anything  else.  The  pound  force 

443914 


38  MECHANICS  AND  HEAT 

is  used  almost  exclusively  as  the  unit  of  force  in  engineering 
work.  Objection  to  its  use  as  a  unit  is  sometimes  made  because 
of  the  fact  that  the  weight  of  a  1-lb.  mass  varies  with  g.  Since 
g  varies  from  978  at  the  equator  to  983.2  at  the  poles  (Sec.  31), 
we  see  that  the  weight  of  a  1-lb.  mass  (or  any  other  mass)  is  about 
1/2  per  cent,  greater  at  the  poles  than  at  the  equator.  This 
slight  variation  in  the  value  of  the  pound  force  may  well  be 
ignored  in  practically  all  engineering  problems.  If  the  standard 
pound  force  is  defined  as  the  weight  of  a  1-lb.  mass  in  latitude  45° 
(g  =  980.6),  it  becomes  as  definite  and  accurate  as  any  other  unit 
of  force. 

33.  Motion  of  Falling  Bodies;  Velocity — Initial,  Final,  and 
Average. — The  initial  velocity  of  a  body  is  usually  represented  by 
v0  (Sec.  26),  and  the  final  velocity  by  vt.  An  example  will  serve 
the  double  purpose  of  illustrating  exactly  what  these  terms  mean 
as  applied  to  falling  bodies,  and  also  of  showing  how  their  numer- 
ical values  are  found. 

Suppose  that  a  body  has  been  falling  for  a  short  time  before  we 
observe  it  and  that  we  wish  to  discuss  its  motion  for  the  succeed- 
ing eight  seconds  of  fall.  Suppose  that  its  initial  velocity  v0, 
observed  at  the  beginning  of  this  eight-second  interval,  is  20  ft. 
per  sec.  Its  final  velocity  vt  at  the  close  of  this  eight-second 
interval  would  be  found  as  follows.  It  will  at  once  be  granted 
that  the  final  velocity  vt  will  be  equal  to  the  initial  velocity  plus  the 
acquired  velocity.  But  by  definition  (Sec.  31),  g  is  numerically 
the  velocity  acquired  or  gained  in  one  second  of  free  fall.  Hence 
in  two  seconds  the  acquired  velocity  would  be  2g,  in  3  seconds  3g, 
and  in  t  seconds  the  velocity  acquired  would  be  gt.  Accordingly 

Vt  =  v0+gt  (10) 

In  the  present  problem  vt  =  20+32.17X8  =  277. 36  ft.  per  sec. 

Average  velocity  is  commonly  represented  by  »  (read  "barred 
v"),  and  in  the  case  of  falling  bodies  it  is  equal  to  half  the  sum  of 
the  initial  and  final  velocities.  Hence 

_    Vo+vt    v0+(vo+gt)  1 

v  =  — ^ — =  ^       '2        =v0+wt  (11) 

In  general,  the  average  velocity  of  a  train  would  not  be  even 
approximately  equal  to  half  the  sum  of  the  initial  and  final 
velocities.  We  ought  therefore  to  prove  the  validity  of  Eq.  11. 


TRANSLATORY  MOTION 


39 


We  readily  see  that  the  average  value  of  all  numbers  from  40  to 
100  is  140-^2  or  70.  If  the  velocity  of  a  train  is  10  feet  per  sec., 
and  each  succeeding  minute  it  gains  2  feet  per  second,  then  its 
velocities  for  the  succeeding  minutes  are  respectively  10,  12,  14, 
16,  18,  20,  22  feet  per  second,  and  its  average  velocity  would 
be,  under  these  special  circumstances,  one-half  the  sum  of 
the  initial  and  final  velocities.  Adding  all  these  numbers  and 
dividing  by  7  gives  an  average  of  16,  but  one-half  the  sum  of  the 
first  and  last  is  also  16. 

We  may  now  make  the  general  statement  that  one-half  the 
sum  of  the  first  and  last  of  a  series 
of  numbers  gives  a  correct  value  for 
the  average,  provided  the  successive 
values  of  the  numbers  in  the  series 
differ  by  a  constant  amount.  Now 
the  velocity  each  successive  second  is 
g  feet  per  second  (approximately  32 
feet  per  second)  greater  than  for  the 
preceding  second;  consequently,  in  all 
cases  of  falling  bodies,  the  average  veloc- 
ity is  half  the  sum  of  the  initial  and 
final  velocities,  as  given  in  Eq.  11. 

The  above  facts  are  shown  graph  i-  \ 

cally  in  Fig.  16,  in  which  the  succes- 
sive lines  1,  2,  3,  4,  5,  .  .  .  t  repre- 
sent the  velocities  of  a  body  after 
falling  1,  2,  3,  4,  .  .  .  t  seconds  re-  pIQ  16 

spectively.     Observe  that  the  velocity 

at  any  time,  e.g.,  after  6  seconds,  consists  of  two  parts;  that 
above  the  horizontal  dotted  line  being  the  initial  velocity  va, 
and  that  below,  the  acquired  velocity  (or  gt),  at  that  instant. 
It  will  be  evident,  as  the  figure  shows,  that  the  average  velocity 
will  be  attained  when  half  of  the  time,  viz.,  4  seconds,  has 
elapsed,  and  hence  v  =  v0+%gt;  whereas  the  final  velocity  vt  is 
attained  after  the  whole  time  t  has  elapsed,  and  is  therefore 
v0+gt,  as  given  above  (Eq.  10). 

In  case  the  body  falls  from  rest,  v0  is  zero,  and  the  conditions 
would  be  represented  by  only  the  portion  of  Fig.  16  below  the 
dotted  line.  In  this  case  the  entire  velocity  vt  at  any  instant 
would  be  merely  gt  or  that  acquired  previous  to  that  instant,  and 
the  average  velocity  1  for  a  given  time  t  would  be  %gt. 


40 


MECHANICS  AND  HEAT 


34.  Distance  Fallen  in  a  Given  Time. — In  general,  the  distance 
d  traversed  by  any  body  in  a  given  time  is  its  average  velocity  v 
times  this  time,  or  d  =  jjt.  Introducing  the  value  of  v  from  Eq. 
11  gives, 

d  =  vt=(v0-\ — n-)t  =  v0t-}-%gt~  (12) 

If  v0  =  Q,  i.e.,  if  the  body  falls  from  rest,  and  the  distance  it  falls 
in  seconds  is  wanted,  then,  from  Eq.  12, 

d  =  $gt2  (13) 

If  v0  =  Q,  Eq.  10  may  be  written  t  =  —.  Substituting  this  value 
of  t  in  Eq.  13,  we  obtain 


vt  =  V2gd  =  V2gh 


(14) 


In  this  equation,  vt  is  the  velocity  acquired  by  a  body  in  falling 
from  rest  through  a  distance  d  (or  k). 

It  will  be  observed  that  v0t  of  Eq.  12  is  the  distance  which  the 
body  with  initial  velocity  v0  would 
travel  in  t  seconds  if  there  were  no 
acceleration;  while  %gt*  is  the  dis- 
tance it  would  travel  in  this  same 
time  if  there  had  been  no  initial  ve- 
locity, i.e.,  had  it  fallen  from  rest. 
The  distance  it  actually  does  travel, 
since  there  are  both  initial  velocity 
and  acceleration,  is  simply  the  sum 
(vector  sum)  of  these  two.  If  a 
person  throws  a  stone  vertically  up- 
ward with  a  velocity  v0,  then  the 
distance  from  that  person's  hand 
to  the  stone  after  t  seconds  will  be 
v0t  —  %  gt2.  For  evidently  the  dis- 
tances the  stone  would  go,  due  to 
its  initial  velocity  alone,  and  due  to 

falling  alone,  are  directly  opposite  as  indicated  by  the  minus 
sign.  Finally,  if  a  person  on  a  high  cliff  throws  a  stone  at  an 
angle  of  45°  (upward)  from  the  horizontal  with  a  velocity  of 
20  ft.  per  second,  let  us  find  the  distance  from  his  hand  to  the 
stone  3  seconds  later.  Due  to  its  initial  velocity  alone,  it  would 
be  60  ft.  distant,  represented  by  line  a  (Fig.  17),  while  due  to 


FIG.  17. 


TRANSLATORY  MOTION  41 

falling  alone  it  would  be  approximately  144  ft.  distant,  repre- 
sented by  line  b.  Hence  due  to  both,  we  have,  by  vector  con- 
struction, HS  (about  100  ft.)  as  the  distance  from  his  hand  to  the 
stone  after  3  seconds  of  its  flight.  The  actual  path  of  the  stone 
is  HCS. 

Distance  Traversed  in  a  Given  Time. — Equations  10,  11,  12,  13, 
and  14,  which  are  derived  from  a  consideration  of  a  particular 
kind  of  uniformly  accelerated  motion,  namely,  that  of  falling 
bodies,  become  perfectly  general  by  substituting  in  them  the 
general  symbol  a  in  place  of  the  particular  symbol  g  to  represent 
the  acceleration.  Making  this  substitution,  these  equations, 
taken  in  order,  become 

vt  =  v0+at  ClOo) 

(lla) 
(12a) 

d  =  \atz  (13a) 

vt=V2ad=V2ah        4*S  *«£-''*      (I4a) 
> 

The  equations  just  given  apply  to  the  motion  of  a  car  when 
coasting  on  a  uniform  grade,  or  to  the  motion  of  any  body  when 
acted  upon  by  a  constant  accelerating  force.  In  the  case  of  a 
car  on  a  uniform  grade,  the  accelerating  force  is,  barring  friction, 
the  component  of  the  car's  weight  which  is  parallel  to  the  grade 
(Fig.  8,  Sec.  19),  and  is  therefore  constant. 

Aside  from  the  motion  resulting  from  gravitational  attraction, 
there  are  very  few  examples  of  uniformly  accelerated  motion. 
Such  motion,  however,  is  very  roughly  approximated  by  many 
bodies  when  starting  from  rest;  e.g.,  by  a  train,  a  steamship,  a 
sailboat,  or  a  street  car.  In  all  these  cases  the  accelerating  force, 
that  is,  the  amount  by  which  the  applied  force  exceeds  friction, 
decreases  rapidly  as  the  speed  increases;  consequently  the  accel- 
eration decreases  rapidly,  and  the  motion  is  then  not  even 
approximately  uniformly  accelerated.  ^ 

35.  Atwood's  Machine. — If  we  attempt  to  make  an  experi- 
mental study  of  the  motion  of  freely  falling  bodies  we  find  that 
the  time  of  fall  must  be  taken  very  small,  or  the  distance  fallen 
will  be  inconveniently  large.  Thus  in  so  short  a  time  as  three 
seconds,  a  body  falls  somewhat  more  than  144  feet.  Hence,  in 
all  devices  for  studying  the  laws  of  falling  bodies  and  verifying 


42 


MECHANICS  AND  HEAT 


experimentally  the  equations  expressing  these  laws,  the  rapidity 
of  the  motion  is  reduced.     Thus  a  wheel  or  a  marble  rolling 
down  an  inclined  plane  experiences  an  acceleration  much  smaller 
than  if  allowed  to  fall  freely.     For  in  the  latter  case  the  acceler- 
ating force  is  the  full   weight   of   the  marble   or  the  wheel; 
while  in  the  former  case  it  is  only  the  component  of  the  weight 
parallel  to  the  incline.     This  reduction  of  the  acceleration  makes 
it  possible  to  study  the  motion  for  a  period  of  several  seconds. 
In  the  Atwood  Machine,  shown  in  its  simplest  form  in  Fig.  18, 
the  reduction  in  the  acceleration  is  attained  in  an  en- 
tirely different  way.     A  and  B  are  two  large  equal 
masses  connected  by  a  light  cord  passing  over  a  light 
wheel  as  shown.     If  a  small   additional  mass   C  is 
placed  on  A,  it  will  cause  A  to  descend  and  B  to  ascend. 
Suppose  that  A  and  B  are  each  150-gm.  masses  and 
that  C  is  a  10-gm.  mass.     If  we  neglect  the  slight  mass 
and  opposing  friction  of  the  wheel,  it  is  clear  that  the 
weight  of  C  is  the  accelerating  force  that  must  accel- 
erate A,  B,  and  C — an  aggregate  mass  equal  to  31 
'  timfcs  the  mass  of  (7;  while  if  C  were  permitted  to  fall 
freely,  its  weight  would  have  to  accelerate  itself  only. 
Hence  the  acceleration  under  these  circumstances  is 
1  1/31  of  that  of  free  fall  or  g,  or  1/31  X980  =  31.6  cm. 
fci  per  sec.  per  sec.,  which  is  about  1  ft.  per  sec.  per 
FIG.  18.    sec.     With  this  value  for  the  acceleration,  we  see 
from  Eq.  13a  that  A  would  "fall"  only  about  4.5 
feet  in  3  seconds.     By  experiment  also  we  find  that  A  "falls" 
4.5  feet  in  3  seconds,  thus  verifying  Eq.  13a. 

The  above  acceleration  may  also  be  calculated  by  means  of  the 
equation  F  =  Ma,  in  which  F  is  the  weight  of  C  in  dynes  and  M  is 
the  combined  mass  of  A,  B,  and  C  in  grams.  A  pendulum  or 
other  device  beating  seconds  is  an  essential  auxiliary.  If  by 
means  of  an  attached  thread,  C  is  removed  after  one  second  of 
"fall,"  A's  velocity,  since  no  accelerating  force  is  then  being 
applied,  will  be  constant,  and  will  have  the  value  31  cm.  per 
sec.  (see  above) ;  while  if  in  another  test  C  remains  3  seconds,  A's 
velocity  at  the  end  of  the  3  seconds  will  be  93  cm.  per  sec.,  as 
may  easily  be  observed.  This  verifies  the  equation  vt  =  at 
(Eq.  7,  Sec.  26). 

36.  Motion  of  Projectiles:  Initial  Velocity  Vertical.— If  a 
rifle  ball  is  fired  vertically  upward,  it  experiences  a  downward 


TRANSLATORY  MOTION  43 

force  (its  weight)  which  slows  it  down,  giving  rise  to  a  negative 
acceleration.  This  decrease  in  velocity  each  second  is  of  course 
32.17  ft.  per  sec.;  so  that  if  the  muzzle  velocity  is  1000  ft.  per  sec., 
the  velocities  after  1,  2,  3,  4,  etc.,  to  t  seconds  are,  respectively, 
1000  —  g  (or  968),  1000  -20, 1000-30, 1000-40  (or  872  ft.  per  sec.), 
etc.,  to  1000  —  gt.  Since  the  velocity  of  the  bullet  is  zero  when 
it  reaches  its  highest  position,  the  number  of  seconds  CO  that 
the  bullet  will  continue  to  rise  is  found  by  placing  1000  —  gt 
equal  to  zero  and  solving  for  t.  CCompare  Sec.  39.)  This  gives 
2  =  31  sec.,  approximately.  The  bullet  requires  just  as  long  to 
fall  back,  so  that  its  time  of  flight  is  62  seconds.  To  get  the 
height  to  which  it  rises,  which  is  obviously  the  distance  it  falls 
in  31  seconds,  let  t  be  31  in  Eq.  13  and  solve  for  d.  We  may  also 
use  the  relation  v  =  \/2gh  (Eq.  14)  to  find  h  if  v  is  known,  or 
vice  versa.  Here  v  =  1000  ft.  per  sec.,  since,  neglecting  air  fric- 
tion, the  bullet,  in  falling,  strikes  the  ground  with  the  same 
velocity  with  which  it  was  fired. 

Throughout  the  discussion  of  projectiles  no  account  will  be 
taken  of  the  effect  of  air  friction,  which  effect  is  quite  pronounced 
on  very  small  projectiles  (Sec.  39).  In  approximate  calcula- 
tions, the  distance  a  body  falls  in  the  first  second  will  be  taken  as 
16  ft.  instead  of  16.08,  and  0  will  be  taken  as  32  instead  of  32.17 
ft.  per  sec.  per  sec.  If  a  rifle  ball  is  fired  vertically  downward, 
e.g.,  from  a  balloon,  with  a  velocity  v0,  its  velocity  will  increase  by 
32  ft.  per  sec.  every  second  (ignoring  air  friction),  so  that  t 
seconds  later  its  velocity  will  be  v0-\-gt.  In  this  case  the  distance 
traversed  in  the  first  t  seconds  is  v0t+^gt2  (Eq.  12);  while  if 
the  initial  velocity  is  upward,  the  distance  from  the  rifle  to  the 
rifle  ball  after  t  seconds  is  v0t  —  %gt2,  as  explained  in  Sec.  34. 

37.  Motion  of  Projectiles:  Initial  Velocity  Horizontal.— If 
a  projectile  is  fired  horizontally,  it  experiences,  the  instant  it 
leaves  the  muzzle  A  of  the  gun  (Fig.  19),  a  downward  pull  (its 
weight)  which  gives  it  a  downward  component  of  velocity  of  32 
ft.  per  sec.  for  every  second  of  flight.  This  causes  it  to  follow  the 
curved  path  AB'C'  .  .  .  F'.  If  it  were  not  for  gravitational 
attraction,  the  bullet  at  the  end  of  the  first,  second,  third,  .  .  . 
etc.,  seconds  would  be  at  the  points  B,  C,  D,  .  .  .  etc.,  respec- 
tively (AB  =  BC  =  CD  =  1000  f t.) ,  instead  of  at  B',  C',  etc.  .  .  . 

To  find  the  velocity  of  the  bullet  at  any  time  t,  say  when  at 
F'  5  sec.  after  leaving  the  muzzle  of  the  gun,  we  simply  find  the 
vector  sum  v'  of  its  initial  velocity  and  its  acquired  velocity,  as 


44  MECHANICS  AND  HEAT 

shown  in  Fig.  19  (left  lower  corner).  The  downward  velocity 
acquired  in  5  sec.  would  of  course  be  gt,  or  160  ft.  per  sec. 
(that  is,  32X5),  and  we  will  assume  1000  ft.  per  sec.  as  the  initial 
horizontal  muzzle  velocity. 

It  will  be  evident  that  the  horizontal  component  of  velocity 
(1000  ft.  per  sec.)  must  be  constant,  for  the  pull  of  gravity  has  no 
horizontal  component  to  either  increase  or  decrease  the  horizontal 
component  of  velocity.  This,  of  course,  is  true  whether  the  initial 
velocity  is  vertical,  horizontal,  or  aslant.  Hence,  neglecting 
friction,  it  is  always  only  the  vertical  component  of  velocity  of  a 
projectile  that  changes. 

To  find  the  distance  that  the  bullet  will  "fall"  in  going  the  first 
1000  ft.,  i.e.,  its  distance  BB'  (Fig.  19)  from  the  horizontal  line 
of  firing  AF,  apply  Eq.  13.  From  this  equation  we  see  that  a 
body  falls  approximately  16  ft.  in  one  second,  64  ft.  in  two  sec., 
and  144  ft.  (i.e.,  16 X32)  in  3  sec.  Hence  55' =  16  ft.,  CC"  =  64ft., 


FIG.  19. 

and  DD'  =  144  ft.,  etc.  To  correct  for  this  falling  of  the  bullet, 
the  rear  sight  is  raised,  causing  the  barrel  to  point  slightly  above 
the  target.  The  greater  the  distance  to  the  target,  the  more  the 
sight  must  be  raised;  the  settings  for  the  different  distances  being 
marked  on  it. 

In  accordance  with  the  above  statements,  it  follows  that  if  a 
bullet  is  dropped  from  a  tower  erected  on  a  level  plain,  and  another 
bullet  is  fired  horizontally  from  the  same  place  at  the  same  in- 
stant, then  the  two  bullets  will  reach  the  ground  at  the  same 
instant,  whether  the  second  one  is  fired  at  a  high  or  low  speed. 
This  fact  can  be  verified  experimentally  (Sec.  40). 

38.  Motion  of  Projectiles :  Initial  Velocity  Inclined. — If  a  rifle 
ball  is  fired  from  a  point  A  (Fig.  20),  in  a  direction  AQ  making  an 
angle  8  with  the  horizontal,  it  describes  a  curved  path  which  may 
be  drawn  as  follows.  Since  distance  is  a  vector,  to  find  where  the 
projectile  will  be  after  a  time  t,  we  simply  obtain  the  vector  sum 
of  the  distance  traversed  in  t  seconds  due  to  its  initial  velocity 
and  the  distance  traversed  in  t  seconds  of  free  fall  from  rest,  as 


TRANSLATORY  MOTION  45 

was  done  in  Sec.  34  (Fig.  17).  Hence  on  the  line  AQ,  which  has 
the  direction  of  the  initial  velocity,  lay  off  the  distances  AB,  BC, 
CD,  DE,  etc.,  each  representing  1000  ft.  (for  a  muzzle  velocity 
of  1000  ft.  per  sec.).  From  B,  C,  D,  E,  etc  ,  draw  the  lines  BB', 
CCr,  DD',  EE',  etc.,  representing  respectively  the  distances  fallen 
in  1,  2,  3,  4,  sec.  Then  here,  just  as  in  Fig.  19,  we  have  BB'  =  16 
ft.,  CC'  =  64  ft.,  DD'  =  U4  ft.,  etc.  The  curve  AB'C'D'E', 
etc.,  represents  the  path  of  the  projectile.  For  consider  any 
point,  e.g.,  K'.  Due  to  its  initial  velocity  alone,  the  projectile 
would  go  from  A  to  K  (10,000  ft.)  in  10  seconds.  Due  to  gravity 
alone  it  would  fall  a  distance  KK',  or  1600  ft.,  in  10  seconds. 
Hence,  due  to  both,  it  covers  the  distance  AK',  the  vector  sum  of 
the  distances  AK  and  KK',  as  shown. 

Note  that  the  straight  line  AK'  gives  not  only  the  magnitude 
but  also  the  direction  of  the  distance  from  A  to  the  projectile 


FIG.  20. 

after  ten  seconds  of  flight.     Note  also  that  AK  is  the  v0t,  and 
that  KK'  is  the  %gtz  of  Eq.  12  (Sec.  34). 

39.  Time  of  Flight  and  Range  of  a  Projectile.— The  Range  is 
the  horizontal  distance  A Q'  (Fig.  20),  or  the  distance  from  the  point  . 
from  which  the  projectile  is  fired  to  the  point  at  which  it  again 
reaches  the  same  level.     The  Time  of  Flight  is  the  time  required 
to  traverse  this  distance. 

To  find  how  long  the  projectile  will  continue  to  rise,  in  other 
words,  to  find  the  time  ti  that  will  elapse  before  its  vertical  com- 
ponent of  velocity  (vv]  will  be  zero,  place  vv  —  gti  =  Q  (i.e.,  ti  =  vv/g 
=  v0  sin  6/g}  and  solve  for  ti  (compare  Sec.  36).  It  was  shown  in 
Sec.  37  that  only  the  vertical  component  of  velocity  changes. 
Since  the  vertical  component  of  velocity  is  zero  at  this  time  ti, 
the  projectile  must  be  at  the  middle  of  its  path  (/',  Fig.  20). 
Therefore  the  time  of  flight. 

T  =  2tt  (15) 


46 


MECHANICS  AND  HEAT 


The  vertical  component  of  velocity  vv  =  v0  sin  6,  and  the  hori- 
zontal component  of  velocity  Vh  =  v0  cos  6  (see  left  upper  corner 
Fig.  20).  If  y0=1000  ft.  per  sec.,  then,  as  the  projectile  leaves 
the  gun,  vv  =  about  240  ft.  per  sec.,  and  VH  =  about  970  ft.  per  sec. 
If  the  angle  6  is  known,  these  two  components  of  the  velocity  may 
be  accurately  found  by  the  use  of  tables  of  sines  and  cosines. 
The  graphical  method  may  also  be  used.  When  £=1  sec.,  i.e., 
1  sec.  after  the  projectile  leaves  the  gun  (see  Fig.  20),  vv  =  208  ft. 
per  sec.  Another  second  later  vv  is  32  ft.  per  sec.  less,  and  when 
t  =  240/32,  or  approximately  8  sec.  after  the  gun  is  fired,  the 
vertical  component  of  velocity  is  zero.  That  is,  in  8  sec.  the  bul- 
let reaches  the  horizontal  part  of  its  path  at  /',  at  which  point 
its  vertical  component  of  velocity  is  clearly  zero.  Since  ti  is 
8  sec.,  the  time  of  flight  T  CEq.  15)  is  16  sec. 

Obviously,  the  range  (R)  is  given  by  the  equation, 

ft  =  VhXT  =  v0  cos  ex2ti  =  v0  cos  9X2v0  sin  B/g         (16) 

Here  the  range  is  15,520  (i.e.,  16  X 970)  ft.  The  Maximum  Height 
reached,  or  /'/",  is  %gtz,  in  which  t  is  the  ti  of  Eq.  15.  For  at 
/'  the  path  is  horizontal,  and  it  was  pointed  out  in  Sec.  37 
that  a  bullet  fired  horizontally  would  reach  the  ground  in  the 
same  time  as  would  a  bullet  dropped  from  the  same  point. 
Hence  I' I"  =  16  X  82  =  1024  ft. 

Effect  of  Air  Friction  on  Velocity  and  Range. — Thus  far,  in  the  study  of 
the  motion  of  projectiles,  we  have  neglected  the  effects  of  air  friction; 
so  that  the  resulting  deductions  apply  strictly  to  a  projectile  traveling 
through  a  space  devoid  of  air  or  any  other  substance,  i.e.,  through  a 
vacuum.  The  theoretical  range  so  found  is  considerably  greater  than  the 
actual  range,  since  the  friction  of  the  air  constantly  decreases  the  veloc- 
ity of  the  projectile  (see  table  below),  and  therefore  causes  it  to  strike 
the  earth  much  sooner  than  it  otherwise  would.  Below  is  given  the  ve- 
locity of  an  Army  Rifle  projectile  in  feet  per  second  at  various  distances 
from  the  muzzle. 


Distance 
in  yds. 

Velocity  in  ft. 
per  sec. 

Distance 
in  yds. 

Velocity  in  ft. 
per  sec. 

100 
200 
300 
400 
600 
800 

1780 
1590 
1420 
1265 
1044 
923 

1000 
1200 
1400 
1600 
1800 
2000 

830 
755 
690 
630 
575 
530 

TRANSLATORY  MOTION  47 

The  angle  (0,  Fig.  20)  which  the  barrel  of  the  gun  makes  with  the 
horizontal  is  called  the  Angle  of  Elevation.  Obviously,  if  the  angle  of 
elevation  is  small,  increasing  it  will  increase  the  range.  It  can  be  shown 
by  the  use  of  calculus  that  the  theoretical  maximum  range  is  obtained 
when  this  angle  is  45°.  The  trigonometric  proof  is  given  below.  For 
heavy  cannon  (12-in.  guns),  the  angle  of  fire  for  maximum  range  is 
nearly  the  same  as  the  theoretical,  namely,  43°;  while  for  the  army  rifle 
it  is  about  31°.  This  difference  is  due  to  the  greater  retarding  effect  of 
air  friction  upon  the  lighter  projectile. 

In  firing  at  targets  1/4  mi.  distant  or  less,  such  as  is  usually  the  case 
in  the  use  of  small  arms,  there  is  not  a  very  marked  difference  between 
the  theoretical  and  the  actual  path  of  the  projectile.  The  maximum 
range  of  the  new  army  rifle  is  about  3  miles.  It  may  be  of  interest  to 
note  that  its  range  in  a  vacuum  (angle  of  elevation  45°)  would  be  about 
24  miles,  and  that  the  bullet  at  the  middle  of  its  flight  would  be  about 
6  miles  above  the  earth,  and  would  strike  the  earth  with  its  original 
muzzle  velocity. 

The  artillery  officer  who  directs  the  firing  at  moving  ships  at  a  distance 
of  5  miles  or  more,  especially  during  a  strong  wind,  must  make  very 
rapid  and  accurate  calculations  or  he  will  make  very  few  "hits."  Many 
other  things  concerning  the  flight"  of  projectiles,  which  are  of  the  utmost 
importance  to  the  artillery  man,  must  be  omitted  in  this  brief  discussion. 

Angle  of  Elevation  for  Maximum  Range. — Since  sin  20  =  2  sin  0  cos  0 
(trigonometry),  Eq.  16  may  be  written 

2sin0cos0_     2sin  20 

g       ~v°  g 

Now  the  maximum  value  of  the  sine  of  an  angle,  namely,  unity,  occurs 
when  the  angle  is  90°.  Therefore  when  20  =  90°,  i.e.,  when  0  =  45°, 
sin  20  is  a  maximum;  hence  the  range  R  is  also  a  maximum,  which  was 
to  be  proved. 

40.  Spring  Gun  Experiment. — From  the  discussion  given  in 
Sec.  38,  it  is  seen  that  if  a  target  at  B,  or  at  C,  or  at  D,  or  at  any 
other  point  on  AQ  (Fig.  20),  is  released  at  the  instant  the  trigger 
is  pulled,  it  will  by  falling  reach  B'  (or  C' ,  or  D',  etc.,  as  the  case 
may  be)  just  in  time  to  be  struck  by  the  bullet.  This  may  be 
shown  experimentally  by  the  use  of  a  spring  gun,  using  wooden 
balls  for  both  projectile  and  target.  The  target  ball  is  held  by  an 
electrical  device  which  automatically  releases  it  just  as  the 
projectile  ball  leaves  the  muzzle  of  the  gun.  The  two  balls  meet 
in  the  air  whether  the  projectile  ball  is  fired  at  a  high  or  low  veloc- 
ity. If  the  target  is  placed  at  the  same  height  as  the  spring  gun, 


48 


MECHANICS  AND  HEAT 


and  the  latter  is  fired  horizontally,  the  two  balls  will  reach  the 
floor  at  the  same  instant. 

41.  The  Plotting  of  Curves. — The  graphical  method  of  presenting 
data  is  found  very  useful  in  all  cases  in  which  a  series  of  several  observa- 
tions of  the  same  phenomenon  has  been  made.  Coordinate  or  cross 
section  paper  is  used  for  this  purpose.  Usually  a  vertical  line  at  the 
left  of  the  page  is  called  the  axis  of  ordinates,  and  a  horizontal  line  at 
the  bottom  of  the  page  is  called  the  axis  of  abscissae.  To  construct  a 
curve,  plot  as  abscissae  the  quantity  that  is  arbitrarily  varied,  and  as 
ordinates  the  corresponding  values  of  the  particular  quantity  that  is 
being  studied.  This  can  be  best  illustrated  by  an  example. 

To  plot  the  results  given  in  the  table,  Sec.  39,  choose  a  suitable  scale 
and  lay  off  200,  400,  etc.,  upon  the  axis  of  abscissae  (Fig.  21)  to  represent 


800         1000         1200 

DISTANCE  IN  YARDS 
FIG.  21. 


1400         1GOO        1300      2000 


the  distance  (from  muzzle  of  gun)  in  yards,  and  400,  800,  etc.,  on  the  axis 
of  ordinates  to  represent  the  velocity  of  the  bullet  in  feet  per  second. 
From  the  table  we  see  that  the  velocity  for  a  range  of  100  yds.  is  1780 
ft.  per  sec.  A  point  A  at  the  center  of  a  small  circle  (Fig.  21)  gives  this 
same  information  graphically,  for  the  abscissa  of  A  is  100  and  its  ordi- 
nate  is  1780.  Point  B,  whose  abscissa  is  200  and  whose  ordinate  is 
1590,  fully  represents  the  second  pair  of  values  (200  and  1590)  in  the  table. 
In  like  manner  the  points  C,  D,  etc.,  are  plotted.  Through  these  points 
a  smooth  curve  is  drawn  as  shown. 

Use  of  the  Curve. — It  will  be  observed  that  the  smooth  curve  passing 
through  all  of  the  other  points  does  not  pass  through  D'.  The  fact  that 
a  point  does  not  fall  on  the  curve  indicates  a  probability  of  error  either 
in  taking  the  data  or  in  plotting  the  results.  In  this  case  a  defective 
cartridge  may  have  been  used  at  the  500-yd.  distance.  A  second  trial 


TRANSLATORY  MOTION  49 

from  that  same  distance  with  a  good  cartridge  would  probably  give  a 
velocity  of  1130  ft.  per  sec.  as  we  would  expect  from  the  curve. 

To  find  the  velocity  at  a  distance  of  900  yds.,  note  that  the  vertical 
line  at  900  strikes  the  curve  at  H.  But  the  ordinate  of  H  is  850.  Hence 
we  know  without  actually  firing  from  that  distance,  that  the  velocity  of 
the  projectile  when  900  yds.  from  the  muzzle  is  850  ft.  per  sec.  This 
method  of  finding  values  is  called  Interpolation.  Such  use  of  curves  for 
detecting  errors  and  for  interpolating  values  makes  them  very  valuable. 
They  also  present  the  data  more  forcibly  than  does  the  tabulated  form, 
for  which  reason  debaters  frequently  use  them.  In  the  physical  labora- 
tory and  in  engineering  work  curves  are  almost  indispensable. 

If  there  were  also  negative  velocities  to  be  plotted,  i.e.,  velocities 
having  a  direction  opposite  to  that  of  the  bullet,  they  would  be  desig- 
nated by  points  at  the  proper  distance  below  the  axis  of  abscissae.  This 
axis  would  then  be  near  the  middle  of  the  coordinate  sheet  instead  of 
at  the  bottom  as  shown. 

42.  Newton's  Three  Laws  of  Motion. — Sir  Isaac  Newton,  the 
great  English  mathematician  and  physicist,  formulated  the  fol- 
lowing fundamental  laws  of  motion  which  bear  his  name. 

1.  A  body  at  rest  remains  at  rest,  and  a  body  in  motion  con- 
tinues to  move  in  the  same  direction  and  at  the  same  speed, 
unless  acted  upon  by  some  external  force. 

2.  The  acceleration  experienced  by  a  given  mass  is  propor- 
tional to  the  applied  force  (accelerating  force),  and  is  hi  the 
direction  of  the  applied  force. 

3.  Action  and  reaction  are  equal,  and  oppositely  directed. 

The  first  law  refers  to  the  inert  character  of  matter,  the  prop- 
erty of  inertia  by  virtue  of  which  any  body  resists  any  change  in 
velocity,  either  in  magnitude  or  direction.  It  is  really  impossible 
to  have  a  body  perfectly  free  from  the  effects  of  all  external 
forces,  but  the  more  we  eliminate  these  effects  by  reducing  fric- 
tion, etc.,  the  more  readily  do  we  observe  the  tendency  of  a  body 
to  keep  in  motion  when  once  started.  The  second  law  states 
the  fact  with  which  we  have  already  become  familiar  in  the  dis- 
cussion of  the  equation  F  =  Ma  (Sec.  25).  The  third  law  is  a 
statement  of  the  fact  that  whenever  and  wherever  a  force  is  applied 
there  arises  an  equal  and  oppositely  directed  force.  This  law  will 
be  further  considered  in  the  next  section. 

43.  Action  and  Reaction,  Inertia  Force,  Principle  of  d'Alem- 
bert. — If  we  press  with  the  hand  upon  the  top,  bottom,  or  side 
of  a  table  with  a  force  of,  say  10  Ibs.,  we  observe  that  the  table 
exerts  a  counter  push  or  force  exactly  equal  to  the  applied  force, 


50  MECHANICS  AND  HEAT 

but  oppositely  directed.  If  the  applied  force  is  increased,  the 
counter  force,  or  Reaction,  is  inevitably  increased.  If,  in  order  to 
push  a  boat  eastward  from  a  bank,  the  oarsman  exerts  a  west- 
ward thrust  (force)  upon  a  projecting  rock  by  means  of  his  oar, 
the  eastward  reacting  thrust  of  the  rock  that  arises  dents  the  oar 
and  starts  the  boat  eastward.  If  an  eastward  pull  is  exerted  on  a 
telephone  pole,  the  guy  wires  to  the  westward  tighten. 

If  a  horse  exerts  a  300-lb.  pull  or  force  FI  upon  the  rope  at- 
tached to  a  canal  boat  a  moment  after  starting,  then  the  backward 
pull  that  the  canal  boat  exerts  upon  the  other  end  of  the  rope 
cannot  possibly  be  either  more  or  less  than  300  Ibs.  Many  peo- 
ple cling  tenaciously  to  the  erroneous  belief  that  the  forward 
pull  of  the  horse  must  be  at  least  slightly  greater  than  the  back- 
ward pull  of  the  boat  or  the  latter  would  not  move.  Many 
people  also  think  that  the  winning  party  in  a  tug-of-war  contest 
must  exert  a  greater  pull  on  the  rope  than  does  the  losing  party, 
which  is  certainly  not  the  case.  For  this  reason,  we  shall  discuss 
very  carefully  the  problem  of  the  horse  and  canal  boat.  The 
applied  force  FI  in  this  case  overcomes  two  forces;  one,  the  fric- 
tion resistance,  say  100  Ibs.,  encountered  by  the  boat  in  moving 
through  the  water,  the  other  (200  Ibs.),  the  backward  pull  exerted 
by  the  boat  because,  by  virtue  of  its  inertia,  it  resists  having  its 
speed  increased.  Note  that  we  are  here  dealing  with  four  forces. 
The  100  Ibs.  of  the  forward  pull  exerted  by  the  horse  just  balances 
the  100-lb.  backward  pull  of  water  friction  on  the  boat;  while  the 
other  200  Ibs.  of  forward  pull  or  force  /i  exerted  by  the  horse, 
just  balances  the  resisting  pull  or  force  /2  that  the  boat  offers  to 
having  its  speed  increased.  Obviously  the  accelerating  f  orce/i  = 
— /2  =  Ma,  in  which  M  is  the  mass  of  the  canal  boat  and  a  is  its 
acceleration.  The  minus  sign  indicates  that  the  forces  are  oppo- 
sitely directed. 

From  this  discussion,  we  arrive  at  the  conclusion  that  the  for- 
ward pull  exerted  upon  any  body  is  exactly  equal  in  magnitude 
to  the  backward  pull  or  resisting  force  exerted  by  the  body.  Thus 
here,  if  the  horse  had  exerted  a  400-lb.  pull,  we  cannot  escape  the 
conclusion  that  the  backward  pull  of  the  boat  would  have  been 
400  Ibs.;  100  Ibs.  being  the  pull  of  water  friction  resistance  as 
before,  and  300  Ibs.  backward  pull  arising  from  the  resistance  the 
boat  offered  to  having  its  speed  increased.  Since  the  accelerating 
force  would  be  300  Ibs.  in  this  case,  instead  of  200  Ibs.  as  before, 
the  acceleration  would  be  1/2  greater  than  before. 


b' 


TRANSLATORY  MOTION  51 

The  above  backward  pull  or  force  that  any  body,  by  virtue 
of  its  inertia,  exerts  in  resisting  change  of  velocity,  has  been  very 
appropriately  called  Inertia  Force.  The  inertia  force  is  always 
numerically  equal  to  the  accelerating  force  that  gives  rise  to  it, 
and  is  always  oppositely  directed.  If  the  canal  boat  were  to  run 
onto  a  sand  bar,  the  friction  would  produce  a  large  negative 
accelerating  force,  and  the  resistance  the  boat  offered  to  decrease 
of  speed  would  develop  an  equal  forward,  or  Driving  Inertia 
Force,  that  would  carry  the  boat  some  distance  onto  the  bar,  even 
though  the  horse  had  ceased  to  pull.  Had  the  sand  bar  been 
more  abrupt,  then  both  the  negative  accelerating  force  and  the 
driving  inertia  force  would  have  been  greater  than  before,  but 
they  would  still  have  been  exactly  equal. 

The  above  fact,  that  all  the  forces  exerted  both  upon  and  by 
any  body  under  any  possible  circumstances  are  balanced  forces, 
i.e.,  that  the  vector  sum  of  all  the  forces  exerted  upon  and  by  a  body 
is  invariably  zero,  is  known  in  mechanics  as  the  Principle  of 
d'Alembert.  In  common  language,  we  frequently  speak  of 
unbalanced  forces.  In  physics,  even,  it  is  frequently  found 
convenient  to  use  the  term,  but  in  such  cases  we  are  simply 
ignoring  the  inertia  force.  Strictly  speaking,  then,  there  is  no 
such  thing  as  unbalanced  forces,  if  all  forces,  including  inertia 
force,  are  taken  into  account.  In  the  above  case  of  the  canal 
boat,  the  only  external  forces  acting  upon  the  boat  to  affect  its 
motion  are  the  forward  pull  exerted  by  the  horse,  and  the  back- 
ward pull  exerted  by  the  water  friction.  These  external  forces 
are  clearly  unbalanced  forces.  In  this  sense,  and  in  this  sense 
only,  may  we  correctly  speak  of  unbalanced  forces. 

44.  Practical  Applications  of  Reaction. — A  horse  cannot  draw 
a  heavy  load  on  a  slippery  road  unless  sharply  shod.  In  order  to 
exert  a  forward  pull  on  the  vehicle,  he  must  exert  a  backward 
push  on  the  ground.  A  train  cannot,  by  applying  the  brakes, 
stop  quickly  on  a  greased  track  because  of  the  inability  of  the 
wheels  to  push  backward  on  the  axle,  and  therefore  on  the  car, 
without  pushing  forward  on  the  track.  The  wheels  cannot, 
however,  exert  much  forward  push  on  a  greasy  rail. 

A  steamship,  by  means  of  its  propellers,  forces  a  stream  of 
water  backward.  The  reaction  on  the  propellers  pushes  the  ship 
forward.  One  of  the  best  suggestions  to  give  a  person  who  is 
learning  to  swim  is  to  tell  him  to  push  the  water  backward.  The 
reaction  forces  the  swimmer  forward. 


52  MECHANICS  AND  HEAT 

An  aeroplane,  by  means  of  its  propellers,  forces  a  stream  of  air 
backward.  The  reaction  on  the  propellers  forces  the  aeroplane 
forward.  The  forward  edge  of  each  plane  or  wing  is  slightly 
higher  than  the  rear  edge.  This  causes  the  planes  to  give  the 
air  a  downward  thrust  as  the  machine  speeds  horizontally  through 
it.  The  reaction  to  this  thrust  lifts  on  the  planes  and  supports 
the  weight  of  the  machine. 

Suppose  that  an  aeroplane,  traveling  50  miles  per  hour,  sud- 
denly enters  a  region  in  which  the  wind  is  blowing  50  miles  per 
hour  in  the  same  direction.  Under  these  circumstances  the  air 
in  contact  with  the  planes,  having  no  horizontal  motion  with 
respect  to  the  planes,  fails  to  give  rise  to  the  upward  reacting 
thrust  just  mentioned,  and  the  aeroplane  suddenly  plunges 
downward.  Such  regions  as  these,  described  by  aeronauts  as 
"holes  in  the  air,"  are  very  dangerous.  It  is  interesting  to  note 
in  this  connection  that  birds  face  the  wind,  if  it  is  blowing  hard, 
both  in  alighting  and  in  starting,  thus  availing  themselves  of  the 
maximum  upward  thrust  of  the  air  through  which  their  wings 
glide. 

45.  Momentum,  Impulse,  Impact,  and  Conservation  of  Momen- 
tum.— The  Momentum  of  a  moving  body  is  denned  as  the  product 
of  the  mass  of  the  body  and  its  velocity,  or 

Momentum  =  Mv  (17) 

The  impulse  of  a  force  is  the  product  of  the  force  and  the  time 
during  which  the  force  acts,  or 

Impulse  =Ft  (18) 

An  impulse  is  a  measure  of  the  ability  of  a  force  to  produce 
motion  or  change  of  motion.  We  readily  see  that  a  force  of  100 
Ibs.  acting  upon  a  boat  for  2  sec.  will  produce  the  same  amount 
of  motion  as  a  force  of  200  Ibs.  acting  for  1  sec.  The  term 
" impulse"  is  usually  applied  only  in  those  cases  in  which  the 
force  acts  for  a  brief  time,  e.g.,  as  in  the  case  of  collision  or 
impact  of  two  bodies,  the  action  of  dynamite  or  powder  in  blast- 
ing, the  firing  of  a  gun,  etc.,  and  the  force  is  then  called  an 
impulsive  force. 

We  shall  now  show  that  an  impulse  is  numerically  equal  to 
the  momentum  change  which  it  produces  in  a  body,  i.e.,  Ft  =  Mv. 
Observe  that  a  "bunted"  ball  loses  momentum  (mainly),  while 
a  batted  ball  loses  momentum  and  then  instantly  acquires  even 


TRANSLATORY  MOTION  53 

greater  momentum  in  the  opposite  direction,  due  to  the  impulse  ap- 
plied by  the  bat.  Obviously,  the  total  change  in  the  momentum 
of  the  ball,  in  case  it  returns  toward  the  pitcher,  is  the  product 
of  the  mass  of  the  ball  and  the  sum  of  its  "pitched"  and  "batted" 
velocities.  If  a  force  F  acts  upon  a  certain  mass  M,  it  imparts 
to  the  mass  an  acceleration,  determined  by  the  equation  F  =  Ma', 
while  if  this  force  acts  for  a  time  t,  the  impulse  Ft  =  Mat.  But 
the  acceleration  of  a  body  multiplied  by  the  time  during  which 
it  is  being  accelerated  gives  the  velocity  acquired.  Hence 

Ft  =  Mat  =  Mv  (19) 

It  should  be  emphasized  that  v  here  represents  the  change  in 
velocity  produced  by  the  impulse  Ft. 

We  shall  next  show  that  when  two  free  bodies  are  acted  upon 
by  an  impulse,  for  example  in  impact  or  when  powder  explodes 
between  them,  then  the  change  of  momentum  of  one  body  is 
exactly  equal  but  opposite  in  sign  to  the  change  in  momentum  of 
the  other.  In  other  words  the  total  momentum  of  both  bodies  is, 
taking  account  of  sign,  exactly  the  same  before  and  after  impact. 
This  law  is  very  appropriately  called  the  law  of  the  Conservation 
of  Momentum. 

Theoretical  Proof  of  the  Conservation  of  Momentum. — Let  us  now 
study  the  effects  of  the  impact  in  a  rear  end  collision,  caused  by 
a  truck  A  of  mass  Ma  and  velocity  va  overtaking  a  truck  B  of 
mass  Mb  and  velocity  Vb.  Let  v'a  and  v'b  be  the  velocities  after 
impact.  During  the  brief  interval  of  impact  t,  truck  A  pushes 
forward  upon  B  with  a  variable  force  whose  average  value  may 
be  designated  by  Fb.  During  this  same  time  t,  truck  B  pushes 
backward  upon  A  with  a  force  equal  at  every  instant  to  the 
forward  push  of  A  upon  B  (action  equals  reaction).  Conse- 
quently the  average  value  Fa  of  this  backward  push  must  equal 
Fb,  and  therefore 

Fbt=-Fat  (20) 

The  minus  sign  in  this  equation  indicates  that  the  forces  are 
oppositely  directed.  In  fact  Fa,  being  a  backward  push,  is 
negative. 

Since  an  impulse  is  equal  to  the  change  in  momentum  which  it 
produces,  and  since  the  change  in  velocity  of  A  is  v'a  —  va,  and  that 
of  B  is  v'b  —  Vb,  we  have 

Fat  =  Ma(v'a  -  va)  and  Fbt  =  Mb(v'b  -  vb) 


54  MECHANICS  AND  HEAT 

Hence,  from  Eq.  20,  we  have 

Mb(v'b-vb]  =  -Ma(v'a-va), 
or 

Mb(v'b-vb}+Ma(v'a-va}=Q  (21) 

From  the  conditions  of  the  problem,  we  see  at  once  that  vb  is  less 
than  v'b,  and  that  va  is  greater  than  v'a.  Accordingly,  in  Eq.  21, 
the  first  term,  which  represents  the  momentum  change  of 
truck  B,  is  positive;  while  the  second  term,  which  represents 
the  momentum  change  of  truck  A  (momentum  lost),  is  negative. 
Since  these  two  changes  are  numerically  equal  but  opposite  in 
sign,  the  combined  momentum  of  A  and  B  is  unchanged  by  the 
impact,  thus  proving  the  Conservation  of  Momentum. 

Observe  in  equation  21  that  the  changes  in  velocity  vary 
inversely  as  the  masses  involved.  Thus  if  B  had  3  times  as 
great  mass  as  A,  its  change  (increase)  in  velocity  would  be  only 
1/3  as  great  as  the  change  (decrease)  in  the  velocity  of  A. 

Briefer  Proof. — The  above  concrete  example  has  been  used  in 
the  proof  for  the  sake  of  the  added  clearness  of  illustration.  We 
are  now  prepared  to  consider  a  briefer,  and  at  the  same  time  more 
general  proof.  In  every  case  of  impact  of  two  bodies,  whatever 
be  their  relative  masses,  or  their  relative  velocities  before  impact, 
the  impulsive  force  acting  on  the  one,  since  action  is  equal  to 
reaction,  is  equal  to,  but  oppositely  directed  to  that  acting  upon 
the  other.  Since  these  two  forces  are  not  only  equal  but  also  act 
for  the  same  length  of  time,  the  two  impulses  are  equal,  and  they 
are  also  oppositely  directed.  But,  since  an  impulse  is  equal  to 
the  change  in  momentum  (Mv)  produced  by  it,  it  follows  that 
the  momentum  changes  of  the  two  bodies  are  equal  but  oppositely 
directed,  and  that  their  sum  is  therefore  zero.  In  other  words, 
the  momentum  before  impact  is  equal  to  the  momentum  after  impact, 
thus  proving  the  Conservation  of  Momentum. 

Experimental  Proof. — Consider  two  ivory  balls  A  and  B  of 
equal  mass  suspended  by  separate  cords  of  equal  length.  Let 
A  be  displaced  through  an  arc  of  say  6  inches  and  then  be  released. 
As  A  strikes  B  it  comes  to  rest  and  B  swings  through  an  equal 
6-inch  arc.  This  shows  that  the  velocity  of  B  immediately  after 
impact  is  equal  to  the  velocity  of  A  immediately  before  impact. 
But  A  and  B  have  equal  mass,  hence  the  total  momentum  is  the 
same  before  and  after  impact,  as  is  required  by  the  law  of  the 
conservation  of  momentum. 


TRANSLATORY  MOTION 


55 


46.  The  Ballistic  Pendulum. — The  ballistic  pendulum  affords 
a  simple  and  accurate  means  of  determining  the  velocity  of  a  rifle 
ball  or  other  projectile.  It  consists  essentially  of  a  heavy  block 
of  wood  P  (Fig.  22),  of  known  mass  M,  suspended  by  a  cord  of 
length  L.  In  practice,  four  suspending  cords  so  arranged  as  to 
prevent  all  rotary  motion  are  used. 

As  the  bullet  b  of  mass  m  and  velocity  v  strikes  P,  it  imparts  to 
P  a  velocity  V  which  causes  it  to  rise  through  the  arc  AB,  thereby 
raising  it  through  the  vertical  height  h.  After  impact,  the  mass 
of  the  pendulum  is  M + m.  From  the  conservation  of  momentum 
we  know  that  the  momentum  of  the  bullet  before  impact,  or  mv, 


'/////////, 


FIG.  22. 

will  be  equal  to  the  momentum  of  the  pendulum  (with  bullet 
embedded)  after  impact,  or  (M+m)V,  i.e., 


(M+m)V 


(22) 


The  values  of  m  and  M  are  found  by  weighing,  and  V  is  found 
from  V=^2gh  (Eq.  14).  For,  as  we  shall  presently  prove,  the 
velocity  which  enables  the  pendulum  to  swing  through  arc  AB, 
or  the  equal  velocity  which  it  attains  in  returning  from  B  to  A, 
is  that  velocity  which  it  would  acquire  in  falling  through  the 
vertical  height  h.  All  other  quantities  being  known,  Eq.  22  may 
then  be  solved  for  v,  the  velocity  of  the  bullet. 

Velocity  Dependent  upon  Vertical  Height  Only. — We  shall  now  show 
that  the  velocity  acquired  by  a  body  in  descending  through  a  given 
vertical  height  h  by  a  frictionless  path,  is  independent  of  the  length  or 
form  of  that  path.  Thus,  if  it  were  not  for  friction,  the  velocity  of  a 
sled  upon  reaching  the  foot  of  a  hill  of  varying  slope  would  be  exactly 
that  velocity  which  a  body  would  acquire  in  falling  through  the  vertical 
height  of  the  hill. 


56  MECHANICS  AND  HEAT 

In  Fig.  22  (upper  right  corner)  let  DE  be  an  incline  whose  slant  height 
is,  say,  four  times  its  vertical  height  DE'  or  h',  i.e.,  DE  =  4h'.  Let  the 
body  C,  starting  from  rest,  slide  without  friction  down  the  incline,  and 
let  C",  also  starting  from  rest,  fall  without  friction.  Let  us  prove  that 
the  velocity  (vt)  of  C  as  it  reaches  E  is  equal  to  the  velocity  (v't)  that  C' 
acquires  in  falling  to  E'.  Note  that  the  vertical  descent  is  the  same 
for  both  bodies. 

The  component  FI  of  C"s  weight  W  is  the  accelerating  force  acting 
upon  C.  From  similar  triangles  we  have 


and    therefore  C"s  acceleration  a  is  0/4.     From  Eq.  14a  we  have  for 
the  velocity  of  C  at  E,  vt=  \/2od  =  \^X4/i=  ^2^  But  from  Eq. 

14  we  have,   for  the  velocity  v't  of  C"  as  it  reaches  E',  v't  =  X/20/j'; 
therefore  Vt  =  v't,  which  was  to  be  proved. 

Further,  it  is  obvious  that  the  same  reasoning  would  apply  had  h'  been 
chosen  larger,  say  equal  to  DF'.  Accordingly,  the  velocity  of  C  upon 
reaching  F,  would  equal  the  velocity  of  C"  upon  reaching  F'.  This  shows 
that  the  increase  in  C"s  velocity  while  going  from  E  to  F  is  equal  to  the 
increase  in  the  velocity  of  C'  in  going  through  the  equal  vertical  distance 
E'F'  (or  EH). 

Let  us  now  consider  the  path  a  b  c  .  .  .  k,  Fig.  22  (lower  right  corner), 
whose  slope  is  not  uniform.  By  subdividing  this  path  into  shorter  and 
shorter  portions,  in  the  limit  each  portion  ab,  be,  cd,  etc.,  would  be  straight, 
and  therefore  abdi,  etc.,  become  triangles  similar  to  triangle  EFH  in  the 
figure  just  discussed.  From  the  discussion  of  triangle  EFH  already 
given,  we  see  that  the  velocities  acquired  by  a  body  in  sliding  without 
friction  through  the  successive  distances  ab,  be,  cd,  etc.,  are  equal  respect- 
ively to  the  velocities  that  would  be  acquired  by  a  body  falling  through 
the  corresponding  successive  distances  hi,  h2,  h3,  etc.  But  the  sum  of 
one  series  is  the  distance  ok,  while  the  sum  of  the  other  series  is  h", 
the  vertical  height  of  ok. 

Consequently  the  total  velocity  acquired  by  a  body  in  sliding 
from  a  to  k,  or  in  general  down  any  frictionless  path,  is  equal  to 
the  velocity  that  would  be  acquired  in  free  fall  through  the  dis- 
tance h",  or  in  general  through  the  vertical  height  of  the  path. 

We  now  see  that  V  of  Eq.  22  is  given  by  the  relation  V  =  ^2gh. 
If  h  is  measured,  V  is  known,  and  therefore  v  of  Eq.  22  is  deter- 
mined. In  practice,  h  is  too  small  to  be  accurately  measured  and 
is  therefore  expressed  in  terms  of  d  and  L  (see  figure)  . 


TRANSLATORY  MOTION  57 


0  PROBLEMS 

1.  The  distance  by  rail  from  a  town  A  to  a  town  B,  120  miles  east  of  A, 
is  240  miles.     The  speed  of  a  train  going  from  A  to  B  is  30  miles  an  hour 
for  the  first  120  miles,  and  20  miles  an  hour  for  the  remainder.     Find  the 
average  speed  and  average  velocity  of  the  train  for  the  run. 

2.  A  train  starts  from  rest  at  a  town  A  and  passes  through  a  town  B 
5.5  miles  to  the  eastward  at  full  speed.     The  excess  pull  upon  the  drawbar 
above  that  required  to  overcome  friction  (i.e.,  the  accelerating  force)  is 
kept  constant,  so  that  the  motion  from  A  to  B  is  uniformly  accelerated. 
The  train  requires  22  minutes  to  make  the  trip.     Find  its  average  velocity 
and  maximum  velocity  in  mi.  per  min.;  mi.  per  hr.;  and  ft.  per  sec. 

3.'  Express  the  acceleration  of  the  train  (Prob.  2)  in  miles  per  hr.  per 
min.;  miles  per  min.  per  min.;  and  feet  per  sec.  per  sec. 

4.1  What  is  the  velocity  of  the  train  (Prob.  2)  15  sec.  after  leaving  A? 
2  min.  after  leaving  A? 

^  5.  How  long  will  it  take  a  2-ton  pull  to  give  a  train  of  40  cars,  weighing 
50  tons  each,  a  velocity  of  1  mi.  per  min.  (i.e.,  88  ft.  per  sec.)  on  a  level 
track?  Neglect  friction. 

6.  Compare  the  intensities  of  illumination  due  to  an  arc  lamp  at  the 
two  distances,  1/2  block,  and  2  blocks. 

7.  A  50-lb.  stone  falls  16  ft.  and  sinks  into  the  earth  1  ft.     Find  its 
negative  acceleration,   assuming  it  to  be  constant  for  this  foot.     Find 
the  force  required  to  penetrate  the  earth.    Suggestion:    Since  the  velocity 
of  the  stone  during  fall  changes  uniformly   from  zero  to  its  "striking" 
velocity,  and  during  its  travel  through  the  earth  from  striking  velocity 
to  zero,  it  follows  that  its  average  velocity  in  air  and  its  average  velocity 
in  earth  are  the  same,  and  that  each  is  equal  to  1/2  the  striking  velocity. 
See  Sec.  33  and  Sec.  45. 

y  8.  If  an  elevator  cable  pulls  upward  with  a  force  of  1200  Ibs.  on  a  1000- 
Ib.  elevator,  what  is  the  upward  acceleration?  How  far  will  it  rise  in  2 
sec.?  Suggestion:  Find  the  accelerating  force  and  express  it  in  poundals, 
not  pounds  (see  Sec.  32).  Neglect  friction. 

9.  How  much  would  a  1 50-lb.  man  weigh  standing  in  the  above  eleva- 
tor if  the  pull  on  the  cable  were  increased  so  as  to  make  the  acceleration 
the  same  as  in  problem  8? 

10.  A  car  that  has  a  velocity  of  64  feet  per  sec.  is  brought  to  rest  in  10 
sec.  by  applying  its  brakes.     Find  its  average  negative  acceleration;  and 
by  comparing  this  acceleration  with  g,  show  graphically  at  what  average 
slant  a  passenger  standing  in  the  car  must  lean  back  during  this  10  sec. 

11.  If  the  car  (Prob.  10)  weighs  30  tons,  what  is  the  forward  push  exerted 
by  its  wheels  upon  the  rails  while  it  is  being  brought  to  rest? 

12.  Prove  that  the  weight  of  a  gram  mass  is  980.6  dynes,  and  that  a 
force  of  1  pound  is  equal  to  32.17  poundals  of  force. 

13.  Reduce  2.5  tons  to  poundals;  to  dynes. 

14.  How  far  does  a  body  travel  in  the  first  second  of  free  fall  from  rest? 
In  the  second?     In  the  third?     In  the  fifth? 

16.  What  is  the  gravitational  pull  of  the  earth  upon  a  mass  of  1  ton  at 
the  moon? 


58  MECHANICS  AND  HEAT 

16.  How  far  will  a  body  fall  in  7  sec.;  and  what  will  be  its  average  and 
final  velocities? 

17.  A  car  on  a  track  inclined  30°  to  the  horizontal  is  released.     How 
far  will  it  travel  in  the  first  7  sec.;  and  what  will  be  its  average  and  final 
velocities  (neglecting  friction)  ?    Compare  results  with  those  of  problem  16. 

18.  How  long  will  it  take  a  body  to  fall  400  meters? 

19.  If  a  rifle  ball  is  fired  downward  from  a  balloon  with  a    muzzle 
velocity  of  20,000  cm.  per  sec.,  how  far  will  it  go  in  4  sec.  ?     If  fired  upward, 
how  far  will  it  go  in  4  sec.? 

20.  A   baseball   thrown   vertically   upward   remains  in  the  air  6  sec. 
How  high  does  it  go?     Observe  that  the  times  of  ascent  and  descent  are 
equal,  neglecting  friction. 

21.  A  stone  is  thrown  upward  from  the  top  of  an  80-ft.  cliff  with  a 
velocity  whose  vertical  component  is  64  ft.  per  sec.'    What  time  will 
elapse  before  it  strikes  the  level  plain  at  the  base  of  the  cliff? 

22.  With  what  velocity  does  a  body  which  has  fallen  2000  ft.  strike  the 
ground? 

23.  A  man  500  ft.  south  of  a  west-bound  train  which  has  a  velocity  of 
60  miles  per  hour,  fires  a  rifle  ball  with  a  muzzle  velocity  of  1000  ft.  per 
sec.  at  a  target  on  the  train.     Assuming  the  aim  to  be  accurate,  how  much 
will  the  bullet  miss  the  mark  if  the  rifle  sight  is  set  for  close  range? 

24.  A  stone  is  dropped  from  a  certain  point  at  the  same  instant  that 
another  stone  is  thrown  vertically  downward  from  the  same  point  with 
a  velocity  of  20  ft.  per  sec      How  far  apart  are  the  two  stones  3  sec.  later? 

25.  A  rifle  ball  is  fired  at  an  angle  of  30°  above  the  horizontal  with  a 
muzzle  velocity  of  1200  ft.  per  sec.    Neglecting  air  friction,  find  the  range 
and  time  of  flight. 

26.  If  the  rifle  ball  (Prob.  25)  is  fired  horizontally  from  the  edge  of  the 
cliff  (Prob.  21),  when  and  where  will  it  strike  the  plain  on  the  level  of  the 
base  of  the  cliff  ? 

27.  If  a  20-ton  car  A,  having  a  velocity  of  5  mi.  an  hr.,  collides  with  and 
is  coupled  to  a  30-ton  car  B  standing  on  the  track,  what  will  be  their  com- 
mon velocity  after  impact? 

28.  If    the  above   car  A  when  it  strikes  B  rebounds  from  it  with  a 
velocity  of  1  mile  per  hour,  find  the  velocity  of  B  after  collision.    Observe 
that  the  total  change  of  A's  velocity  is  6  miles  per  hour.    Will  B'B  change 
be  more  or  less,  and  why? 

29.  A  2-gram  bullet  fired  into  a  2-kilo  ballistic  pendulum  of  length  2 
meters  produces  a  horizontal  displacement  d=10  cm.  (Fig.  22).     Find  the 
velocity  of  the  bullet  in  cm.  per  sec.  and  ft.  per  sec. 


CHAPTER  IV 
ROTARY  MOTION 


47.  Kinds  of  Rotary  Motion. — As  has  previously  been  stated 
(Sec.  22),  a  body  has  pure  rotary  motion  if  a  line  of  particles, 
called  the  axis  of  rotation,  remains  stationary,  and  all  other 
particles  of  the  body  move  in  circular  paths  about  the  axis  as  a 
center.     Familiar  examples  are  the  rotation  of  shafts,  pulleys, 
and  flywheels.     Rotary  motion  is  of  the  greatest  importance  in 
connection  with  machinery  of  all  kinds,  since  it  is  much  more 
common  in  machines  than  reciprocating  motion.     The  study 
of  rotary  motion  is  much  simplified  by  observing  the  striking 
similarity  in  terms  to  those  that  occur  in  the  discussion  of  trans- 
latory  motion. 

Translatory  motion,  as  we  have  seen  (Sec.  22),  may  be  either 
uniform  or  accelerated;  and  the  latter  may  be  either  uniformly  ac- 
celerated or  nonuniformly  accelerated  motion.  The  accelera- 
tion may  also  be  either  positive  or  negative.  Likewise  there 
are  three  kinds  of  rotary  motion:  (a)  uniform  rotary  motion, 
e.g.,  the  motion  of  a  flywheel  or  line  shaft  after  it  has  acquired 
steady  speed;  (6)  nonuniformly  accelerated  motion,  e.g.,  the 
usual  motion  of  a  flywheel  when  the  power  is  first  turned  on  (or 
off);  and  (c)  uniformly  accelerated  rotary  motion,  e.g.,  the 
motion  which  a  flywheel  would  have  if  the  torque  (Sec.  48) 
furnished  in  starting  had  the  proper  value  to  cause  its  increase  of 
rotary  speed  to  be  uniform. 

48.  Torque. — Torque  may  be  defined  as  that  which  produces, 
or  tends  to  produce,  rotary  motion  in  a  body,  just  as  force  is 
that  which  produces,  or  tends  to  produce,  motion  of  translation 
in  a  body.     The  magnitude  of  a  torque  is  force  times  "lever 
arm"  (Eq.25),  and  its  direction  depends  upon  both  the  direction 
and  the  point  of  application  of  the  force.     A  torque  is  not  simply 
a  force,  for  it  is  readily  seen  that  any  force  directed  either 
toward  or  away  from  the  axis,  e.g.,  force  a  (Fig.  23a),  has  no 
tendency  to  produce  rotation.     A  torque  tending  to  produce 
rotation  in  a  counterclockwise  direction  is  called   a  positive 

59 


60 


MECHANICS  AND  HEAT 


torque,   while  a  torque  which  is  oppositely  directed  is  called 
negative. 

Fig.  23a  represents  the  grindstone  shown  in  Fig.  23  as  viewed 
from  a  point  in  line  with  the  axle.  The  torque  due  to  the  force 
a  alone  is  zero.  The  torque  due  to  the  force  b  alone  is  bXOP 
(i.e.,  b.OP),  and  is  negative.  The  torque  due  to  force  c  alone  is 
also  negative,  and  its  magnitude  is  c.OE.  For  the  thrust  c 
equals  the  pull  cf,  which  may  be  thought  of  as  exerted  upon  a 
cord  c'P.  Evidently  the  pull  of  such  a  cord  would  be  just  as 
effective  in  producing  rotation,  at  the  instant  shown,  if  attached 
to  E  on  a  crank  OE,  as  if  attached  to  P  on  the  crank  OP.  Thus 
when  we  define  torque  as  force  times  "lever  arm,"  or 

T  =  Fr  (25) 

we  must  interpret  the  "lever  arm"  r  to  mean  the  perpendicular 
distance  from  the  axis  of  rotation  to  the  line  of  action  of  the  force. 


FIG.  23. 


FIG.  23a. 


The  force  may  be  expressed  in  dynes,  poundals,  pounds,  etc., 
and  the  lever  arm  in  centimeters,  inches,  feet,  etc.;  so  that 
torque  may  be  expressed  in  dyne-centimeter  units,  or  in  poundal- 
feet,  or  pound-feet  units,  etc.  If  several  torques,  some  positive 
and  some  negative,  act  simultaneously  upon  a  flywheel,  the  fly- 
wheel will  start  (or,  if  in  motion,  increase  its  speed)  clockwise, 
provided  the  negative  torques  exceed  the  positive  torques; 
whereas  it  will  start,  or,  if  in  motion,  increase  its  speed  counter- 
clockwise, provided  the  positive  torques  are  the  greater.  If  the 
positive  and  negative  torques  just  "balance,"  then  the  fly- 
wheel will  remain  at  rest;  or  if  already  in  motion,  its  speed  will 
not  change. 


ROTARY  MOTION  61 


77Ae  Couple. — Two  equal  and  oppositely  directed  forces 
which  do  not  have  the  same  line  of  action  (F  and/'1',  upper  sketch, 
Fig.  24)  constitute  a  Couple.  The  torque  developed  by  this 
couple  is  equal  to  the  product  of  one  of  the  forces,  and  the  dis- 
tance AC  between  them,  and  is  entirely  independent  of  the  posi- 
tion (in  the  plane  of  the  figure)  of  the  pivot  point  about  which 
the  body  rotates.  The  torque  due  to  this  particular  couple  is 
also  counterclockwise  (positive)  whether  the  pivot  point  is  at 
A,  B,  C,  D,  or  at  any  other  point.  If  A  is  the  pivot  point,  then 
the  force  F  produces  no  torque,  while  Ff  produces  the  positive 
torque  F'XAC  (i.e.,  F'.AC).  If  B  is  the  pivot  point,  then  both 
forces  produce  positive  torques;  but, 
since  the  lever  arm  for  each  is  then 
only  \  AC,  the  total  torque  is  the 
same  as  before.  If  D  is  the  pivot 
point,  then  F'  produces  a  negative 
torque,  and  F,  a  positive  torque;  but, 
since  F  acts  upon  a  lever  arm  which 
is  longer  than  that  of  F'  by  the  dis- 
tance AC,  it  follows  that  the  sum  of 
these  two  torques  about  D  is  F.AC  as 
before,  and  is  also  positive. 

If  three  men  A,  B,  and  C  by  pushing  Fio.  24. 

with  one  hand  and  pulling  with  the 

other  apply  respectively  upon  the  wheel  E  (Fig.  24)  the  cou- 
ples represented  by  FI  and  Ft,  F3  and  F4,  and  F6  and  F6,  then 
each  man  will  contribute  an  equal  positive  torque  helping  to 
rotate  the  wheel.  For,  as  sketched,  the  forces  are  all  equal, 
and  the  distances  a,  b,  and  c  are  equal;  consequently  the  three 
torques  are  equal.  But  from  the  above  discussion  we  see  that 
the  torques  due  to  these  three  similar  couples  will  be  equal  about 
any  point  in  the  plane  of  the  wheel,  and  hence  about  its  axis. 

49.  Resultant  Torque,  and  Antiresultant  Torque. — Let  the 
forces  a,  b,  c,  and  d,  Fig.  23a,  be  respectively  20,  12,  14,  and  40 
pounds,  and  let  OP  =  1  ft.,  OE  =  8  in.,  and  OF  =  4  in.  The  torque 
due  to  a  is  zero;  that  due  to  b  is  12X1  or  12  lb.-ft.,  or  144  lb.-in., 
negative;  the  torque  due  to  c  is  14X8  or  112  lb.-in.,  negative,  and 
that  due  to  d  is  40  X4  =  160  lb.-in.  positive.  The  sum  of  all  these 
torques,  that  is  the  one  torque  that  would  be  just  as  effective  in 
producing  rotation  as  all  of  these  torques  acting  simultaneously, 
is  96  lb.-in.  or  8  lb.-ft.,  a  negative  torque.  Consequently,  one 


62  MECHANICS  AND  HEAT 

force,  say  h,  acting  in  the  direction  6,  but  of  magnitude  8  Ibs., 
would  produce  just  as  great  a  torque  as  would  all  four  forces, 
a,  b,  c,  and  d  acting  together.  This  torque  may  be  called  the 
Resultant  of  the  other  four  torques.  If  the  force  h  is  reversed 
in  direction,  it  produces  a  positive  torque  of  8  lb.-ft.,  called  the 
Antiresultant  torque.  This  antiresultant  torque,  acting  with  the 
torques  due  to  a,  b,  c,  and  d,  would  produce  equilibrium.  Ob- 
viously, this  antiresultant  torque,  instead  of  being  an  8-lb.  force 
on  a  1-ft.  arm,  might,  for  example,  be  a  4-lb.  force  on  a  2-ft.  arm, 
or  a  16-lb.  force  on  a  6-in.  arm. 

50.  Angular  Measurement.  —  Angles  may  be  measured  in 
degrees,  minutes,  and  seconds,  in  revolutions,  or  in  radians.  In 
circular  measure,  an  angle  is  found  by  dividing  the  subtended 
arc  by  the  radius,  that  is, 


If  the  arc  equals  the  radius,  then  the  angle  is  of  course  unity,  and 
is  called  one  Radian.     Thus  angle  AOC  (Fig.  25)  is  one  radian 
because  arc  ABC  equals  the  radius  r.     The 
angle  AOB,  or  8,  is  1/2  radian  because  the 
arc  AB  is  1/2  the  radius  r.     Since  the  cir- 
cumference of  a  circle  is  2nr,  it  follows  that 
there  are  2ir  radians  in  360°,  or  the  radian 
equals  57°.3.     In  the  study  of  Mechanics, 
angles,  angular  velocity,  and  angular  ac- 
FIG.  25.  celeration  are  almost  always  expressed  in 

terms  of  radians  instead  of  degrees. 

61.  Angular  Velocity  and  Angular  Acceleration.  —  Angular 
velocity  is  the  angle  traversed  divided  by  the  time  required;  or, 
since  the  unit  of  time  is  usually  the  second,  it  is  numerically  the 
angle  turned  through  in  one  second.  If  a  certain  flywheel  makes 
600  revolutions  per  min.  (written  600  R.P.M.),  its  angular 
velocity 

w  =  10  rev.  per  sec.,  or  62.8  (i.e.,  10X27r)  radians  per  sec. 

If  the  rotary  speed  of  the  flywheel  is  constant  during  the  one 
minute,  the  above  62.8  radians  per  sec.  is  its  actual  angular 
velocity  at  any  time  during  that  minute;  whereas  if  its  speed 
fluctuates,  then  62.8  radians  per  sec.  is  simply  the  average  angular 
velocity  co  (read  "barred  omega")  for  the  minute  considered. 


ROTARY  MOTION  63 

Again,  suppose  that  the  above  flywheel,  starting  from  rest  and 
uniformly  increasing  its  speed,  makes  600  revolutions  in  the  first 
minute.  Its  average  angular  velocity  w  is  62.8  rad.  per  sec.  as 
before;  but,  since  its  initial  velocity  is  zero,  its  angular  velocity 
ojt  at  the  close  of  the  first  minute  must  be  twice  the  average,  or 
125.6  rad.  per  sec.  (Compare  v  of  Sec.  33.)  Since  this  angular 
velocity  is  acquired  in  one  minute,  the  angular  acceleration  (a)  is 
given  by  the  equation 

CO* 

a  =  —  =  125.6  radians  per  sec.  per  min. 

In  one  second  the  wheel  will  acquire  1/60  as  much  angular  veloc- 
ity as  it  does  in  1  min.;  hence  we  may  also  write 

a  =  2.09  radians  per  sec.  per  sec. 

which  means  that  in  one  second  the  increase  in  angular  velocity 
is  2.09  radians  per  sec.  Evidently,  at  a  time  t  seconds  after 
starting,  the  angular  velocity  <at=od.  Thus  5  seconds  after 
starting  co  =  10.45  radians  per  sec. 

To  summarize  (see  also  Sees.  52  and  57),  we  have,  in  transla- 
tory  motion, 

distance  traversed  d 

Average  velocity  -       time  required     >  or  V  =  T 

In  rotary  motion 

angle  traversed        _      0 
Average  angular  velocity  =   ^time  required  >or"  =  J    (27> 

.        gain  in  velocity  •»•  — » 

Acceleration  (trans,  motion)  =  —r. —       — ^rm  or  a 


time  required  '  t 

gain  in  angl.  velocity  co,  —  co0  ,, 

Angular  aceelerat.on  =         time  r°quired p*.  or  «- -y-  (28) 

52.  Relation  between  Linear  and  Angular  Velocity  and 
Acceleration. — If,  due  to  a  constant  accelerating  torque,  a  body 
starts  from  rest  with  a  constant  angular  acceleration  a,  and,  in  a 
time  t,  rotates  through  an  angle  6  and  acquires  an  angular  veloc- 
ity w,  then  it  will  be  true  that  any  mass  particle  in  this  body  at  a 
distance  r  from  the  axis  travels,  in  this  time  t ,  a  distance  d  =  rd 
(note  that  arc  =  r0,  Eq.  26),  acquires  in  this  time  a  linear  velocity 
v  =  rw,  and  experiences  during  this  same  time  a  linear  accelera- 
tion a  =  ra. 

Proof:  Dividing  both  sides  of  the  equation  d  =  r6  by  t,  gives 

. 


64  MECHANICS  AND  HEAT 

r\ 

v  =  r-.  =  ru.    If  a  body  starts  from  rest  with  uniform  acceleration, 

its  average  velocity  v  is  of  course  only  half  as  great  as  its  final 
velocity  v;  hence  v  =  2v.  Likewise  w  =  2co.  Hence,  since  v  =  ro>, 
it  follows  that  v  =  rw.  Now  a  =  v/t;  therefore,  dividing  both 
sides  of  the  equation  v  =  rco  by  t,  gives  a  =  ru/t  =  ra.  Accordingly 

d  =  rd,    v  =  rw,  and  a  =  ra  (29) 

If  6  is  given  in  radians,  co  in  radians  per  second,  and  a  in 
radians  per  second  per  second,  then  if  r  is  given  in  feet,  d  will  be 
expressed  in  feet,  v  in  feet  per  second,  and  a  in  feet  per  second  per 
second.  From  Eq.  29,  we  see  (1)  that  the  distance  which  a  belt 
travels  is  equal  to  the  product  of  the  radius  (?•)  of  the  belt  wheel 
over  which  it  passes,  and  the  angle  6  (in  radians)  through  which 
this  wheel  turns;  (2)  that  the  linear  velocity  of  the  belt  is  equal 
to  r  times  the  angular  velocity  of  the  wheel  in  radians  per  second, 
and  (3)  that  the  linear  acceleration  which  the  belt  experiences  in 
starting,  is  equal  to  r  times  the  angular  acceleration  of  the  belt 
wheel  expressed  in  radians  per  second  per  second. 

Let  it  be  required  to  find  the  angular  velocity  co  of  the  drivers 
of  a  locomotive  when  traveling  with  a  known  velocity  v.  From 
Eq.  29  we  have  co  =  y/r;  hence,  dividing  the  linear  velocity  of  the 
locomotive  expressed  in  feet  per  second  by  the  radius  of  the  driver 
in  feet,  we  obtain  co  in  radians  per  second. 

53.  The  Two  Conditions  of  Equilibrium  of  a  Rigid  Body.— 
If  the  resultant  of  all  of  the  forces  acting  upon  a  body  is  zero, 
the  First  Condition  of  Equilibrium  is  satisfied  (Sec.  17),  and  the 
body  will  remain  at  rest,  if  at  rest,  or  continue  in  uniform  motion 
in  a  straight  line  if  already  in  motion.  If,  in  addition,  the  result- 
ant of  the  torques  acting  upon  the  body  is  zero,  the  Second  Con- 
dition of  Equilibrium  is  satisfied,  and  the  body  will  remain  at  rest, 
if  at  rest,  or  if  already  rotating  its  angular  velocity  will  neither 
increase  nor  decrease.  Forces  which  satisfy  the  first  condition 
of  equilibrium  may  not  satisfy  the  second.  The  general  case  of 
several  forces  acting  upon  various  points  of  the  body,  and  in 
directions  which  do  not  all  lie  in  the  same  plane,  is  too  complex 
to  discuss  here.  The  simpler  but  important  case  of  three  forces 
all  lying  in  the  same  plane  will  now  be  considered. 

A  body  acted  upon  by  three  forces  which  lie  in  the  same  plane 
is  in  equilibrium  if  (a)  the  three  forces  when  represented  graphic- 
ally form  a  closed  triangle  (first  condition  of  equilibrium);  and 


ROTARY  MOTION  65 

(&)  if  the  lines  of  action  of  these  three  forces  meet  in  a  point 
(second  condition  of  equilibrium).  Thus  the  body  A  (Fig.  26) 
is  in  equilibrium,  since  the  three  forces  a,  b,  and  c,  form  a  closed 
triangle  as  shown,  and  they  also  (extended  if  necessary)  meet 
at  the  point  E. 

The  three  forces  a',  &',  and  c'  which  act  upon  the  body  B  (Fig. 
27),  when  graphically  represented  form  a  closed  triangle  and 
therefore  have  zero  resultant.  Consequently  they  have  no  tend- 
ency to  produce  motion  of  translation  in  the  body,  but  they  do 
tend  to  produce  rotation.  For  the  forces  b'  and  c'  meet  at  D, 
about  which  point  the  remaining  force  a'  clearly  exerts  a  clock- 
wise torque;  hence  the  second  condition  of  equilibrium  is  not 
fulfilled. 

That  forces  a,  b,  and  c  (Fig.  26)  produce  no  torque  about  E  is 


FIG.  26.  FIG.  27. 

evident,  since  all  three  act  directly  away  from  E.  It  may  not 
be  equally  evident  that  they  produce  no  torque  about  any  other 
point  in  A,  such  as  F.  That  such  is  the  case,  however,  may  be 
easily  shown.  The  two  forces  a  and  b  have  a  resultant,  say  c", 
which  is  equal  to  c  but  oppositely  directed  (since  the  three  force 
a,  6,  and  c  are  in  equilibrium);  hence  a  and  b  may  be  replaced 
by  c"  acting  downward  at  E.  But  obviously  c  and  c"  would 
produce  equal  and  opposite  torques  about  F,  or  about  any  other 
point  that  may  be  chosen.  Hence  three  forces  which  form  a  closed 
triangle  and  also  meet  in  a  point  have  no  tendency  to  produce  either 
translation  or  rotation  of  a  body. 

Applications  to  Problems. — A  ladder  resting  upon  the  ground  at 
the  point  A  (Fig.  28)  and  leaning  against  a  frictionless  wall  at  B, 
supports  at  its  middle  point  a  200-lb.  man  whose  weight  is 


66 


MECHANICS  AND  HEAT 


represented  by  W.  Neglecting  the  weight  of  the  ladder,  let  us 
find  the  thrusts  a  and  b.  Since  the  ladder  is  in  equilibrium,  the 
three  forces  a,  b,  and  W  which  act  upon  it  must  meet  at  a  point 
and  must  also  form  a  closed  triangle.  The  thrust  b  must  be 
horizontal,  since  the  wall  is  frictionless,  and  it  therefore  meets 
W  produced  at  C.  The  upward  thrust  of  the  ground  on  the 
ladder  must  also  pass  (when  extended)  through  C;  i.e.,  it  must 
have  the  direction  AC.  To  find  the  magnitude  of  a  and  of  6, 
draw  W  to  &  suitable  scale,  and  through  one  end  of  W  draw  a  line 
parallel  to  6,  and  through  the  other  end  draw  a  line  parallel  to  a. 
The  intersection  of  these  two  lines  determines  the  magnitude  of 
a  and  of  6,  as  explained  under  Fig.  7, 
Sec.  18. 

Since  the  crane  beam  in  the  problem 
at  the  close  of  this  chapter  is  acted 
upon  by  three  forces,  and  since  it  is 
also  in  equilibrium,  the  problem  may 
be  solved  by  the  method  of  this  sec- 
W  tion. 

54.  Moment  of  Inertia  and  Accel- 
erating Torque. — The  mass  of  a  body 
may  be  defined  as  that  property  by  vir- 
tue of  which  the  body  resists  a  force 
tending  to  make  it  change  its  velocity. 
The  Moment  of  Inertia  of  a  body,  e.g.,  of  a  flywheel,  is  that  property 
by  virtue  of  which  the  flywheel  resists  a  torque  tending  to  make 
it  change  its  angular  velocity.  Consider  a  steam  engine  which  is 
belted  to  a  flywheel  connected  with  a  buzz  saw,  as  in  the  case  of  a 
small  saw  mill.  The  difference  between  the  tension  on  the  tight 
belt  and  the  slack  belt,  times  the  radius  of  the  pulley  over  which 
the  belt  passes,  gives  the  applied  torque.  If  the  applied  torque 
is  just  sufficient  to  overcome  the  opposing  torque  due  to  friction  of 
bearings,  and  the  friction  encountered  by  the  saw,  then  the  speed 
remains  uniform;  while  if  the  applied  torque  exceeds  this  value, 
the  angular  velocity  increases,  and  its  rate  of  increase,  that  is,  the 
angular  acceleration,  is  proportional  to  this  excess  torque.  If  the 
applied  torque  is  less  than  the  resisting  torque,  the  angular  accel- 
eration is  negative,  that  is,  the  flywheel  slows  down,  and  the 
rate  at  which  it  slows  down  is  proportional  to  the  deficiency  in 
torque.  Compare  with  accelerating  force,  Sec.  25. 

The  relation  between  the  moment  of  inertia  7  of  a  flywheel,  th<3 


28. 


ROTARY  MOTION  67 

accelerating  torque,  and  the  resulting  angular  acceleration  a,  is 
given  by  the  following  equation, 

Accelerating  torque  =  la,  i.e.,  T  =  Ia  (30) 

Compare  with  F  =  Ma  (Eq.  5,  Sec.  25).  If  we  apply  a  known 
torque  and  determine  a  experimentally,  we  may  find  the  numer- 
ical value  of  I  from  Eq.  30.  If  the  torque  is  expressed  in  dyne- 
centimeters  (i.e.,  the  force  in  dynes  and  the  lever  arm  in  centi- 
meters) and  a  in  radians  per  sec.  per  sec.,  then  I  will  be.expressed 
in  C.G.S.  units  (see  also  Sec.  55).  From  Eq.  30  we  see  that  unit 
torque  will  give  a  body  of  unit  moment  of  inertia  unit  angular 
acceleration;  while  from  Eq.  5,  we  see  that  unit  force  will  give 
unit  mass  unit  linear  acceleration. 

The  moment  of  inertia  of  two  similar  wheels  is  found  to  be 
proportional  to  the  products  of  the  mass  and  the  radius  squared 
for  each  (Eq.  31,  Sec.  55).  Hence  we  find  fly 
wheels  made  with  large  mass  and  large  radius, 
and  with  the  greater  part  of  the  mass  in  the 
rim,  for  which  part  the  radius  is  largest. 

65.  Value  and  Unit  of  Moment  of  Inertia.  — 
We  shall  now  determine  the  relation  between 
the  C.G.S.  unit  of  moment  of  inertia  (Sec. 
54)  and  the  mass  and  radius  of  the  rotating 
body,  say  a  wheel.  We  shall  first  determine  FIG.  29. 

the  expression  for  the  moment  of  inertia  of  a 
particle  of  mass  mi  at  a  distance  r\  from  the  axis  of  rotation 
(Fig.  29).  Let  us  consider  only  this  mass  mi,  ignoring,  for  the 
time  being,  the  mass  of  the  rest  of  the  wheel.  To  further  sim- 
plify the  discussion,  let  the  force  F\  that  produces  the  accelerating 
torque  T\,  act  upon  m\  itself,  so  that 


miari 

AirM,    -jj\crC 
since  a  =  ria  (Eq.  29,  Sec.  52). 

But  this  same  accelerating  torque  =  /ia  (Eq.  30),  in  which  I\ 
is  the  moment  of  inertia  of  nil  about  the  axis  through  0,  and  a 
is  its  angular  acceleration  about  the  same  axis. 
Consequently 


or  /i  =  Wxri2  (31) 


68  MECHANICS  AND  HEAT 

Likewise,  the  moment  of  inertia  72  of  ra2  (see  Fig.  29)  can  be 
shown  to  be  ra2r22,  and  that  of  m3  to  be  W3r32,  etc.  Now  if  we 
add  together  the  moments  of  inertia  of  all  the  mass  particles  of 
the  wheel  we  have  for  the  moment  of  inertia  of  the  entire  wheel 


This  may  be  briefly  written 

I=2mr2  (32) 

in  which  Zrar2  (read  sigma  mr2)  signifies  a  summation  of  rar2  for 
all  of  the  mass  particles  in  the  wheel. 

If,  in  Eq.  31,  all  quantities  are  expressed  in  C.G.S.  units, 
then  m  will  be  expressed  in  grams,  r  in  centimeters,  and  hence 
7  will  be  expressed  in  gm.-cm.2  units.  If  units  of  the  F.P.S. 
system  are  used,  7  will  be  expressed  in  lb.-ft.2  units.  Thus  a 
2000-lb.  flywheel  having  practically  all  of  its  mass  in  the  rim  of 
mean  radius  5  feet,  would  have  a  moment  of  inertia  I  =  Mr2 
(approx.)  =  50,000  (i.e.,  2000  X52)  lb.-ft.2  For  the  r  of  Eq.  32 
is  practically  the  same  (i.e.,  5  ft.)  for  every  mass  particle  in  the 
wheel,  and  the  combined  mass  of  all  these  particles  is  M  or 
2000  Ibs. 

The  moment  of  inertia  of  an  emery  wheel  or  grindstone  of 
radius  r  and  mass  M  is  obviously  less  than  Mr2;  for  in  this  case 
the  mass  is  not  mainly  concentrated  in  the  "rim,"  since  many  of 
the  mass  particles  move  in  circles  of  very  small  radius  r.  It  can 
be  shown  by  the  use  of  higher  mathematics  that  the  moment  of 
inertia  of  such  disc-like  bodies  is 


(33) 
For  a  sphere  of  radius  r  and  mass  M 

I  =  %Mr2  (34) 

66.  Use  of  the  Flywheel.  —  The  purpose  of  a  flywheel,  in 
general,  is  to  "steady"  the  motion.  Thus,  in  the  above- 
mentioned  case  of  the  saw  mill  (Sec.  54),  if  the  applied  torque 
furnished  by  the  steam  engine  is  greater  than  all  the  resisting 
friction  torques,  this  excess  torque,  or  accelerating  torque, 
causes  the  speed  of  the  flywheel  to  increase;  while  if  the  saw 
strikes  a  tough  knot,  so  that  the  friction  torques  exceed  the 
applied  torque,  then  the  flywheel  helps  the  engine  to  run  the  saw, 


ROTARY  MOTION  69 

and  in  so  doing  is  slowed  down.  Indeed  the  flywheel,  when  its 
speed  is  increasing,  is  storing  up  energy,  which  is  again  handed 
on  to  the  saw  when  its  speed  decreases. 

It  is  a  matter  of  common  observation  that  a  heavy  wheel, 
when  being  set  in  motion  with  the  hand,  offers  an  opposing 
backward  inertia  torque;  while  if  we  attempt  to  slow  down  its 
motion,  it  offers  an  opposing  forward  inertia  torque,  or  Driving 
Inertia  Torque.  It  is  just  this  driving  inertia  torque,  developed 
by  the  flywheel  when  slowing  down,  that  helps  the  engine  to  run 
the  saw  through  the  tough  knot.  Compare  this  with  the  driving 
inertia  force  that  pushes  the  canal  boat  onto  the  sand  bar  (Sec. 
43).  If  one  were  to  shell  some  corn  with  the  ordinary  hand  corn 
sheller,  both  with  and  without  the  flywheel  attached,  he  would  be 
very  forcibly  impressed  with  the  fact  that,  at  times,  the  flywheel 
assists  with  a  driving  torque. 

In  the  case  of  "four  cycle"  gas  engines  (Chap.  XVIII)  which 
have  one  working  stroke  to  three  idle  strokes  (i.  e.,  the  three 
strokes  during  which  the  gas  is  not  pushing  upon  the  piston), 
it  must  be  clear  that  the  flywheel  runs  not  only  the  machinery, 
but  also  the  engine  itself,  during  these  three  strokes.  Doing 
this  work,  i.e.,  supplying  the  driving  torque  during  the  three 
idle  strokes,  necessarily  slows  down  the  flywheel,  but  this  lost 
speed  is  regained  during  the  next  stroke,  or  working  stroke, 
when  the  explosion  occurs.  If  the  flywheel  is  too  light,  this 
fluctuation  in  speed  is  objectionably  great.  Since,  in  the  case 
of  steam  engines,  every  stroke  is  a  working  stroke,  lighter  fly- 
wheels suffice  than  for  gas  engines  of  the  same  horse  power  and 
speed. 

The  flywheel  of  a  high  speed  gas  engine  need  not  have  so 
great  moment  of  inertia  as  is  required  for  a  lower  speed  engine 
furnishing  the  same  horse  power.  In  each  case,  to  be  sure,  the 
flywheel  "carries"  the  load  during  the  three  idle  strokes,  but 
the  time  for  these  three  idle  strokes  is  shorter  for  the  high  speed 
engine.  (Flywheel  design  will  be  considered  in  Sec.  76). 

57.  Formulas  for  Translatory  and  Rotary  Motion  Compared. 
— Below  will  be  found  a  collection  of  formulas  applied  to 
translatory  motion,  and  opposite  them  the  corresponding  for- 
mulas for  rotary  motion.  The  similarities  and  differences  in 
these  two  sets  of  formulas  should  be  observed.  All  of  these 
formulas  should  be  thoroughly  understood,  and  most  of  them 
may  be  memorized  with  profit. 


70  MECHANICS  AND  HEAT 

Translatory  Motion  Rotary  Motion 
d  6 

-•-T 

vt  =  at  or,  v0-\-at  ut= 

-v  =v0+%at  w  = 

N      Vt  —V0      Vt  Wj—  co°        wt 

a(oTg)  =  —  j  —  OTJ  a=  —  ^  —  ory 

d  =  v~t  B  =  ut 

d  =t*  9  =  %at2 


F  =Ma  T  =Ia 

F  is  accelerating  force.  T  is  accelerating  torque. 

Kinetic  energy  =  \Mvi  Kinetic  energy  =  |/co2 

(Energy  is  discussed  in  Chap.  VL.) 

PROBLEMS 

1.  Reduce  2.5  revolutions  (a)  to  radians;  (6)  to  degrees.     Express  the 
angle  between  north  and  northeast  in  (c)  radians;  (d)  degrees  J  and  (e) 
revolutions. 

2.  A  shaft  makes  1800  R.P.M.     Find  «  in  radians  per  sec.;  in  degrees 
per  sec. 

3.  Through  how  many  degrees  will  a  shaft  rotate  in  3  min.,  if  w=20 
radians  per  sec.? 

4.  A  flywheel,  starting  from  rest  with  uniformly  accelerated  angular 
motion,  makes  15  revolutions  in  the  first  10  sec.     What  is  its  average 
angular  velocity  (a)  in  revolutions  per  sec.?     (b)  In  radians  per  sec.? 
(c)  In  degrees  per  sec.?     (d)  What  is  its  velocity  at  the  close  of  the  first 
10  sec.? 

6.  What  is  the  angular  acceleration  of  the  flywheel  of  (problem  4)  (a)  in 
radians  per  sec.  per  sec.?     (b)  In  radians  per  sec.  per  min.? 

6.  A  belt  which  travels  at  the  rate  of  30  ft.  per  sec.  drives  a  pulley  whose 
radius  is  3  in.     What  is  the  angular  velocity  for  the  pulley  ? 

7.  A  small  emery  wheel  acquires  full  speed  (1800  R.P.M.)  5  sec.  after 
starting.     Assuming  the  angular  acceleration  to  be  constant,   find  its 
value  for  this  5  sec. 

8.  Through  what  angle  does  the  emery  wheel  rotate  (Prob.  7)  in  the 
first  5  sec.? 

9.  Find  the  total  torque  produced  by  the  forces  a,  b,  c,  and  d  (Sec.  49)  if 
o  and  b  are  both  reversed  in  direction. 

10.  A  locomotive  has  a  velocity  of  30  miles  per  hr.  one  minute  after 
starting,     (a)  What  is  its   average   acceleration  for   this   minute?     (b) 
What  is  the  average  angular  acceleration  of  its  drivers,  which  are  6  ft.  in 
diameter? 

11.  A  crane  (Fig.  7,   Sec.  18)  is  lifting  a  load  of  2400  Ibs.     Find  the 
thrust  of  the  beam  B  against  the  post  A  and  the  pull  on  cable  C  due  to  this 


ROTARY  MOTION 


71 


load,  if  B  is  30  ft.  in  length  and  inclines  30°  to  the  vertical,  and  if  C  is 
attached  to  B  at  a  point  10  feet  from  O,  and  to  A  at  a  point  20  feet 
above  the  foot  of  B.  Use  the  graphical  method  and  compare  with  the 
ladder  problem,  Sec.  53. 

12.  The  arms  AO  and  OB  of  the  bell  crank  (Fig.  30)  are  equal.     Find 
the  pull  F,  and  also  the  thrust  of  0  on  its  bearings. 

13.  Find  the  required  pull  and  thrust  (Prob.  12)  if  F  has  the  direction  BC. 
Compare  ladder  problem,  Sec.  53. 


14.  The  belt  which  drives  a  1600-lb.  flywheel,  whose  rim  has  an  average 
radius  of  2  feet  (assume  mass  to  be  all  in  the  rim),  passes  over  a  pulley  of 
1-ft.  radius  on  the  same  shaft  as  the  flywheel.  The  average  pull  of  the 
tight  belt  exceeds  that  of  the  slack  belt  by  100  Ibs.  Neglecting  friction, 
how  long  will  it  take  the  flywheel  to  acquire  a  velocity  of  600  R.P.M. 
First  find  /,  then  a,  etc. 


CHAPTER  V 

UNIFORM  CIRCULAR  MOTION,  SIMPLE  HARMONIC 
MOTION 

58.  Central  and  Centrifugal  Forces,  and  Radial  Acceleration. — 
If  a  body  moves  in  a  circular  path  with  uniform  speed,  it  is  said 
to  have  Uniform  Circular  Motion.  If  a  stone,  held  by  a  string, 
is  whirled  round  and  round  in  a  horizontal  circular  path,  it  has 
approximately  uniform  circular  motion.  In  order  to  compel 
the  stone  to  follow  the  curved  path,  a  certain  inward  pull  must  be 
exerted  upon  the  string  by  the  hand.  This  pull  is  termed  the 
Centripetal  or  Central  force.  The  opposing  pull  or  force  exerted 
by  the  stone  by  virtue  of  its  inertia  (which  inertia  in  accordance 
with  Newton's  first  law  tends  to  make  it  move  in  a  straight  line 
tangent  to  the  circle),  is  exactly  equal  to  this  central  force  in  mag- 
nitude, and  is  termed  the  Centrifugal  force. 

If  the  string  breaks,  both  the  central  and  centrifugal  forces 
disappear,  and  the  stone  flies  off  in  a  straight  line  tangent  to  its 
path  at  that  instant.  The  pull  upon  the  string  causes  the  stone 
to  change  its  velocity  (not  in  magnitude  but  in  direction)  and  is 
therefore  an  accelerating  force  and  equal  to  Ma,  in  which  M  is 
the  mass  of  the  stone  and  a,  its  acceleration.  Hence  to  find  the 
pull  upon  the  string  it  will  be  necessary  to  weigh  the  stone  to  get 
M,  and  also  to  compute  the  acceleration  a.  Observe  that  the 
applied  accelerating  force  is  the  pull  of  the  string;  while  the  cen- 
trifugal force  is  really  the  inertia  force  that  arises  due  to  the  resist- 
ance the  stone  offers  to  having  its  velocity  changed  (in  direction). 

Here,  as  in  all  possible  cases  that  may  arise,  the  accelerating 
force  and  the  inertia  force  are  equal  in  magnitude  but  oppositely 
directed,  and  they  disappear  simultaneously  (Sec.  43) .  The  simul- 
taneous disappearance  of  the  central  and  centrifugal  forces  at  the 
instant  the  string  breaks,  is  in  complete  accord  with  the  behavior 
of  all  reactions.  Thus,  so  long  as  we  push  down  upon  a  table, 
we  experience  the  upward  reacting  thrust;  but  the  instant  we 
cease  to  push,  the  reacting  thrust  disappears. 

Centrifugal  force  has  many  important  applications,  for 
example,  in  the  cream  separator  (Sec.  60),  the  centrifugal  gov- 

72 


UNIFORM  CIRCULAR  MOTION 


73 


ernor  (Sec.  63),  the  centrifugal  pump,  and  centrifugal  blower 
(Sec.  150).  It  is  this  force  which  causes  too  rapidly  revolving 
flywheels  and  emery  wheels  to  "burst"  (Sec.  59),  and  it  is  also 
this  force  which  necessitates  the  raising  of  the  outer  rail  on  curves 
in  a  railroad  track  (Sec.  62).  The  centrifugal  clothes  dryer 
used  in  laundries,  and  the  machine  for  separating  molasses  from 
.sugar,  used  in  sugar  refineries,  both  operate  by  virtue  of  this 
principle.  The  centrifugal  force  due  to  the  velocity  of  the  earth 
in  its  orbit  prevents  the  earth  from  "falling"  to  the  sun  (Sec. 
29),  while  the  centrifugal  force  due  to  its  rotation  about  its  axis 
causes  the  earth  to  flatten  slightly  at  the  poles  and  bulge  at  the 


"^     . 


FIG.  31. 

equator.     The  polar  diameter  is  about  27  miles  less  than  the 
equatorial  diameter. 

To  find  the  Radial  Acceleration  a,  construct  a  circle  (Fig.  31) 
whose  radius  r  represents  the  length  of  the  string.  Let  S  repre- 
sent the  stone  at  a  certain  instant  (£  =  0),  at  which  instant  it  is 
moving  west  with  a  velocity  v0.  After  a  time  t  (here  t  is  chosen 
about  1/2  sec.),  the  stone  is  at  Si,  and  its  velocity  vt  is  the  same 
in  magnitude  as  before,  but  is  directed  slightly  south  of  west. 
Its  velocity  has  evidently  changed,  and  if  this  change  is  divided 
by  the  time  t  in  which  the  change  occurred,  the  result  is  by 
definition  (Sec.  24)  the  acceleration  a. 


74  MECHANICS  AND  HEAT 

This  change  in  velocity,  or  the  velocity  acquired,  is  readily 
found  by  drawing  from  S  (Fig.  31,  upper  sketch)  two  vectors, 
SA  and  SB,  to  represent  v0  and  vt  respectively,  and  then  con- 
necting A  and  B.  Obviously  the  acquired  velocity  is  that  velocity 
which  added  (vectorially)  to  v0  gives  vt;  consequently  it  is  repre- 
sented by  the  line  AB.  Acquired  velocity,  however,  is  always 
given  by  the  product  of  acceleration  and  time,  or  at;  hence, 
the  velocity  AB  =  at. 

The  triangles  OSSi  and  SAB  are  similar,  since  their  sides  are 
perpendicular  each  to  each;  and  if  6  is  very  small,  arc  SS\  may  be 
considered  equal  to  chord  SSi.  But  SSi  is  the  distance  the  stone 
travels  in  the  time  t,  or  v0t.  Hence,  from  similar  triangles, 

at    v0t  v0z 


Since  F  =  Ma,  the  central  force,  usually  designated  as  Fc,  is 
given  by  the  equation 

F,=^  (36) 

As  already  stated,  the  centrifugal  force  and  the  central  force  are 
equal  in  magnitude  but  oppositely  directed,  hence,  Fc  (Eq.  36) 
may  stand  for  either.  If  M  is  the  mass  of  the  stone  S  in  pounds, 
v  its  velocity  in  feet  per  second,  and  r  is  the  length  of  the  string 
in  feet;  then  Fc  is  the  pull  on  the  string  in  poundals,  not  pounds 
(see  Sees.  25  and  32).  If  Mis  given  in  grams,  v  in  centimeters 
per  second,  and  r  in  centimeters,  then  Fc  is  the  pull  in  dynes, 
not  grams  of  force.  By  means  of  this  equation  we  may  compute 
the  forces  brought  into  play  in  the  operation  of  the  centrifugal 
clothes  dryer,  cream  separator,  steam  engine  governor,  or  in  the 
case  of  a  fast  train  rounding  a  curve. 

In  many  cases  it  is  found  more  convenient  to  use  a  formula 
involving  angular  velocity  in  revolutions  per  second  instead  of 
linear  velocity.  If  a  body,  e.g.  a  wheel,  makes  n  revolutions  per 
second,  its  ''rim"  velocity,  or  the  distance  traversed  in  one  second 
by  a  point  on  the  rim  of  the  wheel,  is  n  circumferences  or  2irrn. 
Substituting  this  value  for  v  in  Eq.  36  we  have 

(37) 


For,  since  one  revolution  is  2r  radians,  w  =  2,irn,  and  o>2  = 


. 

UNIFORM  CIRCULAR  MOTION  75 

Central  Force  Radial. — That  Fc  is  radial  is  apparent  in  the 
above  case,  since  the  force  must  act  in  the  direction  of  the  string. 
That  this  is  equally  true  in  the  case  of  a  flywheel  or  cream  separa- 
tor, or  in  all  cases  of  uniform  circular  motion,  may  be  seen  from 
a  discussion  of  Fig.  32.  For  if  the  central  force  Fc  acting  upon  a 
particle  P  which  is  moving  to  the  left  in  the  circle,  had  the  direc- 
tion a,  there  would  be  a  component  of  this  force,  a',  acting  in  the 
direction  of  the  motion,  and  hence  tending  to  increase  the 
velocity;  if,  on  the  other  hand,  Fc  acted  in  the  direction  b,  there 
would  be  a  component  of  the  force,  b',  acting  in  such  a  direction 
as  to  decrease  the  velocity.  But  if 
P  has  uniform  circular  motion,  its 
velocity  must  neither  increase  nor 
decrease;  hence  neither  of  these  com- 
ponents, a'  and  &',  can  be  present,  and 
Fc  must  therefore  be  radial. 

That  the  acceleration  is  radial  can 
be  shown  in  another  way.     As  point 
Si  (Fig.  31)  is  taken  closer  and  closer 
to  S  (i.e.,  as  t  is  chosen  smaller  and 
smaller)  vt  becomes  more  nearly  par- 
allel to  va,  and  AB  (see  upper  sketch,  JPIQ-  32. 
Fig.  31)    becomes  more   nearly  per- 
pendicular   to    v0.     In    the    limit,    as    Si    approaches    S,  AB 
becomes  perpendicular  to  v0,  and  therefore  parallel  to  r.    But 
the  acceleration  has  the  direction  AB,  hence  it  is  radial.     It 
should  also  be  emphasized  that  the  acceleration  is  linear  (not 
angular),  and  is  therefore  usually  expressed  either  in  feet  per 
second  per  second  or  in  centimeters  per  second  per  second 
(Sec.  24). 

69.  Bursting  of  Emery  Wheels  and  Flywheels. — The  central 
force  Fc  required  to  cause  the  material  near  the  rim  of  a  revolving 
emery  wheel  to  follow  its  circular  path,  is  usually  enormous. 
If  the  speed  is  increased  until  Fc  becomes  greater  than  the  strength 
of  the  material  can  withstand,  then  the  material  pulls  apart, 
and  we  say  that  the  emery  wheel  "bursts."  It  is  evident  that 
it  does  not  burst  in  the  same  sense  that  it  would  if  a  charge  of 
powder  were  exploded  at  its  center.  In  the  latter  case  the 
particles  would  fly  off  radially;  while  in  the  former  they  fly  off 
tangentially.  Indeed,  the  instant  the  material  cracks  so  that 
the  central  force  disappears,  the  centrifugal  force  also  disappears 


76 


MECHANICS  AND  HEAT 


(Sec.  58),  and  each  piece  moves  off  in  a  straight  line  in  the 
direction  in  which  it  happens  to  be  moving  at  that  instant. 

60.  The  Cream  Separator.— The  essentials  of  a  cream  separator 
are,  a  bowl  A  (Fig.  33),  attached  to  a  shaft  B,  and  surrounded  by 
two  stationary  jackets  C  and  E.     When  B  is  rapidly  revolved 
by  means  of  the  "worm"  gear,  as  shown,  the  fresh  milk,  enter- 
ing at  G,  soon  acquires  the  rotary  motion  of  the  bowl,  and,  due 
to  its  inertia  which  tends  to  make  it  move  in  a  straight  line,  it 
crowds  toward  the  outside  of  the  bowl  with  a  force  Fc. 

Both  the  cream  and  the  milk  particles  tend  to  crowd  outward 
from  the  center  of  the  bowl,  but  the  milk  particles  being  heavier 
than  cream  particles  of  the  same  size, 
experience  the  greater  force,  and  a  sepa- 
ration takes  place.  In  the  figure  the 
cross-hatched  portion  c  represents  the 
cream,  and  the  space  between  this  and 
the  bowl,  marked  m,  represents  the 
milk.  Small  holes  marked  a  permit  the 
cream  to  fly  outward  into  the  stationary 
jacket  E,  from  which  it  flows  through 
the  tube  F  into  the  cream  receptacle,  the 
holes  marked  6,  farther  from  the  center 
of  the  bowl  than  holes  a,  permit  the 
skim-milk  to  fly  outward  into  the  sta- 
tionary jacket  C,  from  which  it  flows 
through  the  tube  D  into  the  milk  re- 
ceptacle. 

The  bowls  of  many  of  the  commercial 

separators  contain  numerous  separating  chambers  designed  to 
make  them  more  effective.  This  simple  form,  however,  illus- 
trates the  features  common  to  all.  With  a  good  cream  sepa- 
rator about  98  or  99  per  cent,  of  the  butter  fat  is  obtained;  i.e., 
1  to  2  per  cent,  remains  in  the  skim-milk.  In  the  case  of 
"cold  setting"  or  gravity  separation  and  skimming  as  usually 
practised,  5  per  cent,  or  more  remains  in  the  skim-milk. 

61.  Efficiency  of  Cream  Separator. — In  fresh  milk,  the  cream  is 
distributed  throughout  the  liquid  in  the  form  of  finely  divided  par- 
ticles.    If  allowed  to  stand  for  several  hours  the  cream  particles, 
being  slightly  lighter  than  the  milk  particles,  slowly  rise  to  the  sur- 
face.    Thus  a  separation  of  the  cream  from  the  milk  takes  place, 
and,  since  it  is  due  to  gravitational  force,  it  is  termed  "gravita- 


FIG 


UNIFORM  CIRCULAR  MOTION  77 

tional"  separation.  Calling  the  mass  of  one  of  these  cream  par- 
ticles mi  and  the  mass  of  an  equal  volume  of  milk  m»,  the  pull  of 
the  earth  (in  dynes,  Sec.  32)  on  the  cream  particles  is  m\g  and 
the  pull  on  the  milk  particles  is  m^g.  The  difference  between  these 
two  pulls,  m2g  —  mig,  or  g(m?.—m\)  constitutes  the  separating  force. 
This  slight  separating  force  is  sufficient  to  cause  the  cream  par- 
ticle to  travel  from  the  bottom  of  a  vessel  to  the  top,  a  distance  of 
one  foot  or  so  in  the  course  of  a  few  hours. 

In  the  case  of  the  centrifugal  separator,  the  force  with  which  mz 
crowds  toward  the  outside  of  the  bowl  is  47r2w2rm2  (Eq.  37)  while 
for  the  cream  particle  it  is  4ir2n2rmi.  The  difference  between  <  \ 
these  two  forces,  or  4?r2n2r  (mz  —  mi),  is,  of  course,  the  separating 
force  which  causes  the  cream  particle  to  travel  toward  the  center. 
The  ratio  of  this  separating  force  to  the  separating  force  in  the 
case  of  gravity  separation  is  sometimes  called  the  separator  effi- 
ciency. Hence 


, 
Efficiency  =  —  —         ---     ~  (38) 


In  the  above  equation,  if  the  gram  and  the  centimeter  are  used 
throughout  as  units  of  mass  and  length  respectively,  the  separat- 
ing force  will  be  expressed  in  dynes;  while  if  pounds  and  feet  are 
used,  the  force  is  expressed  in  poundals,  not  pounds  (see  Sec.  58). 
The  word  efficiency  is  used  in  several  distinctly  different  ways  — 
the  more  usual  meaning  brought  out  in  Sec.  85,  being  quite 
different  from  that  here  given. 

62.  Elevation  of  the  Outer  Rail  on  Curves  in  a  Railroad  Track. 
—  Let  B  (Fig.  34)  represent  a  curve  in  the  railroad  track  ABC. 
Suppose  that  for  a  short  distance  this  curve  is  practically  a  circle 
of  radius  ri  with  center  of  curvature  at  E.  Let  it  be  required  to 
find  the  "proper  elevation"  d  of  the  outer  rail  in  order  that  a  car, 
when  passing  that  particular  part  of  the  curve  with  a  velocity  Vi, 
shall  press  squarely  against  the  track,  so  that  its  "weight,"  so- 
called,  shall  rest  equally  on  both  rails. 

On  a  level,  straight  track,  the  thrust  of  the  car  against  the 
track  is  simply  the  weight  of  the  car,  and  is  vertical;  whereas  on 
a  curve,  the  thrust  T7!  (lower  sketch,  Fig.  34),  is  the  resultant  of 
the  weight  of  the  car  W  and  the  centrifugal  force  Fc  which  the 
car  develops  in  rounding  the  curve  (Eq.  36,  Sec.  58).  These 
forces  should  all  be  considered  as  acting  on  the  center  of  mass 
0  of  the  car  (Sec.  95).  If  the  velocity  of  the  car  is  such  that 


78 


MECHANICS  AND  HEAT 


Mvi2/r  (i.e.,  Fc),  has  the  value  shown,  then  the  total  thrust  T\ 
will  be  perpendicular  to  the  track,  and  consequently  the  thrust 
will  be  the  same  on  both  rails. 

If  the  car  were  to  pass  the  curve  at  a  velocity  twice  as  great  as 
that  just  mentioned  (or  2z>0,  the  centrifugal  force  would  be  quad- 
rupled, and  would  therefore  be  represented  by  the  line  OH. 
This  force,  combined  with  W,  would  give  a  resultant  thrust  7" 
directed  toward  the  outer  rail.  The  inner  rail  would  then  bear 
no  weight,  while  the  thrust  Tr  on  the  outer  rail  would  be  about 
one-half  greater  than  the  entire  weight  W  of  the  car,  as  the  figure 


FIG.  34. 


shows.  The  least  further  increase  in  velocity  would  cause  the 
car  to  overturn.  This  theoretical  limiting  velocity  could  never  be 
reached  in  practice,  because  either  the  wheel  flanges  or  the  rail 
would  give  way  under  the  enormous  sidewise  thrust.  Indeed 
whenever  the  above-mentioned  velocity  v\,  which  may  be  called 
the  "proper"  velocity,  is  exceeded,  the  wheel  flanges  push  out  on 
the  outer  rail.  If  this  sidewise  push  is  excessive,  a  defective 


flange  may  give  way  and  cause  a  wreck, 
velocity  should  not  be  much  exceeded. 


Hence  the  "proper' 


UNIFORM  CIRCULAR  MOTION  79 

From  the  figure  it  may  be  seen  that 

From  the  figure  we  also  have 

-fi  =  sin  0i,  or  di  =  D  sin  0i  (40) 

Observe  that  the  two  angles  marked  0i  are  equal  (sides  perpen- 
dicular each  to  each).  Knowing  the  values  of  v\t  g,  and  r\  we 
may  determine  tan  0i  from  Eq.  39.  Having  found  the  value  of 
tan  0i,  we  may  obtain  0i  by  the  use  of  a  table  of  tangents.  If 
the  width  of  the  track  D  is  also  known,  the  proper  elevation  di 
of  the  outer  rail  may  be  found  from  Eq.  40.  All  quantities  in- 
volved in  Eqs.  39  and  40  must  be  expressed  either  in  F.P.S. 
units  throughout,  or  else  in  C.G.S.  units  throughout.  Observe 
that  for  radius  r\  we  use  v\,  T\,  Q\,  and  di  respectively  for  the 
"proper"  velocity,  thrust,  angle,  and  elevation. 

In  practice  the  curvature  is  not  made  uniform,  but  decreases 
gradually  on  both  sides  of  the  place  of  greatest  curvature  until 
the  track  becomes  straight;  while  the  elevation  of  the  outer  rail 
likewise  gradually  decreases  until  it  becomes  zero,  where  the 
straight  track  is  reached.  This  construction  eliminates  the 
violent  lurching  of  the  car,  which  would  occur  if  the  transition 
from  the  straight  track  to  the  circular  curve  were  sudden. 

63.  The  Centrifugal  Governor. — The  essential  features  of  the 
centrifugal  governor,  or  Watt's  governor,  used  on  steam  engines, 
are  shown  in  the  simplest  form  in 
Fig.  35.  The  vertical  shaft  S, 
which  is  driven  by  the  steam  engine, 
has  attached  to  its  upper  end  two 
arms  c  and  d  supporting  the  two 
metal  balls  A  and  B  as  shown. 

It  will  readily  be  seen  that  the 
weight  of  the  balls  tends  to  bring 
them  nearer  to  the  shaft,  while  the 
centrifugal  force  tends  to  make  FIQ>  35. 

them  move  farther  from  the  shaft. 

If,  then,  the  speed  of  the  engine  becomes  slightly  greater  than 
normal,  the  balls  move  farther  out,  c  and  d  rise  (see  dotted  posi- 
tion), and  by  means  of  rods  e  and  /  cause  collar  C  to  rise.  By 


80  MECHANICS  AND  HEAT 

means  of  suitable  connecting  levers,  this  upward  motion  of  C 
partially  closes  the  throttle  valve.  The  supply  of  steam  being 
reduced,  the  speed  of  the  engine  drops  to  normal.  If,  on  the 
other  hand,  due  to  a  sudden  increase  in  load,  the  speed  of  the 
engine  drops  below  normal,  then  the  balls,  arms  c  and  d,  and 
collar  (7,  all  lower.  This  lowering  of  C  opens  the  throttle  wider 
than  normal,  thereby  supplying  more  steam  to  the  engine,  and 
restoring  the  normal  speed. 

In  some  engines,  when  the  speed  becomes  too  low,  the  governor 
automatically  adjusts  the  inlet  valve  so  that  steam  is  admitted 
during  a  greater  fraction  of  the  stroke.  This  raises  the  average 
steam  pressure  on  the  piston,  and  the  normal  speed  is  regained. 
This  subject  will  be  further  considered  under  "cut  off  point" 
in  the  chapter  on  the  steam  engine. 

63a.  The  Gyroscope. — Just  as  a  body  in  linear  motion  resists  change 
in  direction  (Sec.  58),  so  a  rotating  flywheel  resists  any  change  in  direction 
of  rotation,  i.e.,  it  resists  any  shifting  in  direction  of  its  axis.  By  virtue 
of  this  principle,  a  rapidly  rotating  flywheel,  properly  mounted,  will 
greatly  reduce  the  rolling  of  a  ship,  as  has  been  shown  by  tests.  This 
principle  may  also  yet  be  successfully  applied  in  securing  greater  stability 
for  aeroplanes. 

The  Gyroscope  in  its  simplest  form  is  shown  in  Fig.  35a.  This  device, 
until  recent  years,  was  merely  an  interesting,  perplexing  scientific  toy. 
The  wheel  W  rotates  as  indicated  by  arrow  c  at  a  high  speed  and  with 
very  little  friction  on  the  axle  AB.  If,  now,  the  end  of  the  axle  A  is 
rested  upon  the  supporting  point  P,  the  end  B,  which  is  without  support, 
does  not  drop  in  the  direction  d  as  it  would  if  the  wheel  were  not  rotat- 
ing, but  moves  horizontally  round  and  round  the  point  of  support  as 
indicated  by  arrow  e. 

The  mathematical  treatment  of  the  gyroscope  is  very  difficult;  so  that 
we  shall  here  simply  state  a  few  facts  with  regard  to  its  motion.  The 
angular  velocity  w  of  the  wheel  W  is  a  vector,  and  may  be  represented 
at  a  given  instant  by  the  arrow  co0,  called  a  rotor.  Observe  that  if  W 
were  a  right-handed  screw,  a  rotation  in  the  direction  indicated  by  arrow 
c  would  advance  the  screw  in  the  direction  of  arrow  w0. 

Following  this  same  convention  we  see  that  the  torque  produced  by  the 
weight  of  the  wheel  would  tend  to  produce  rotation,  i.e.,  would  produce 
an  angular  acceleration  about  the  horizontal  axis  indicated  by  the  rotor 
a,  and  further  that  the  direction  of  this  angular  acceleration  would  be 
properly  represented  by  placing  the  arrow  head  on  the  end  of  a  away 
from  the  reader.  Note  that  rotor  a  lies  in  the  axis  of  torque.  The 
rotation  of  B  in  the  direction  of  arrow  e  (horizontal)  with  a  constant 
angular  velocity  «'  about  a  vertical  axis  through  P,  is,  by  this  same  con- 


UNIFORM  CIRCULAR  MOTION  81 

vention  (right-handed  screw)  properly  represented  by  the  rotor  «'. 
From  the  figure,  we  see  that  w»  lies  in  the  axis  of  spin,  and  a in  the 
axis  of  torque.  The  vertical  axis  in  which  lies  a'  is  called  the  Axis  of 
Precession.  The  change  in  direction  of  the  axis  AB  is  called  Precession. 

As  an  aid  to  the  memory,  using  the  right  hand,  place  the  middle  finger 
at  right  angles  to  the  forefinger  and  the  thumb  at  right  angles  to  both. 
Next  point  the  forefinger  in  the  direction  of  rotor  ua  and  the  middle  finger 
in  the  direction  of  a.  It  will  then  be  found  that  the  thumb  points  in 
the  direction  of  the  rotor  «'  (i.e.,  down,  not  up). 

Cause  of  Precession. — Since  rotors  are  vectors,  they  may  be  added 
graphically.  In  the  figure,  «<,  represents  the  angular  velocity  of  W  at 
a  given  instant.  Its  angular  velocity  ««  a  short  time  t  later  would  be 
given  by  the  equation  ut=u0+(»a,  in  which  w0  is  the  angular  velocity 
acquired  in  the  short  time  t.  But  angular  velocity  acquired  (gained) 
is  at  (Eq.  28,  Sec.  51).  Therefore  ut  =  w0+ at  as  shown  graphically  in 
Fig.  35a,  in  which  the  rotor  at  is  drawn  from  the  arrow  point  of  rotor  w0. 
The  resultant  is  the  closing  side,  or  the  new  angular  velocity  w«,  which 


FIG.  35a. 

differs  from  «„  only  in  direction.  In  other  words,  during  this  short  time 
t,  the  axis  has  changed  in  direction  through  the  angle  0,  and  B  has 
moved  in  the  direction  e.  Clearly  6  is  the  angle  of  precession  in  this 
time  t,  and  e/t  is  the  precessional  angular  velocity  w'. 

Compare  this  change  in  direction  (not  in  magnitude)  of  rotary  motion 
with  the  change  in  direction  of  linear  motion  (centrifugal  force,  Sec.  58). 
Observe  in  the  vector  diagram  given  in  Fig.  35a  that  u0,  at,  and  ut 
correspond  respectively  to  v0,  at,  and  vt  of  the  vector  diagram  shown  in 
Fig.  31. 

The  Reeling  (precession  of  axis  of  rotation)  of  a  top  when  its  axis  is 
inclined,  is  due  to  this  gyroscopic  action.  In  fact  if  B  is  considerably 
higher  than  A  (Fig.  35a)  when  A  is  placed  uponP,  the  gyroscope  becomes 
essentially  a  reeling  top. 

Due  to  the  rotation  of  the  earth  (centrifugal  force),  the  equatorial 
diameter  is  27  miles  greater  than  the  polar  diameter.  Since  the  axis  of 
the  earth  inclines  to  the  normal  to  the  plane  of  its  orbit  around  the  sun 
("Plane  of  the  Ecliptic")  by  an  angle  of  23°.5,  the  gravitational  pull  of 


82  MECHANICS  AND  HEAT 

the  sun  (and  also  the  moon)  on  this  equatorial  protuberance  produces  a 
torque  about  an  axis  perpendicular  to  the  earth's  axis  of  spin,  just  as 
the  weight  of  the  top  (when  inclined)  produces  a  torque  about  an  axis 
lying  on  the  floor  and  at  right  angles  to  the  spindle  of  the  top  (axis  of 
spin). 

Thus  thf  earth  reels  like  a  great  top  once  in  about  26,000  years.  The 
earth's  axis  if  extended  would  sweep,  each  26,000  years,  around  a  circle 
of  23°. 5  radius  with  a  point  in  the  sky  in  the  direction  normal  to  the  plane 
of  the  ecliptic  as  center  of  this  circle.  Consequently,  13,000  years 
from  now  the  earth's  axis  will  point  in  a  direction  47°  from  our  present 
pole  star,  Polaris.  This  reeling  of  the  earth  causes  the  Precession  of  the 
Equinoxes  around  the  ecliptic  once  in  26,000  years. 

Monorail  Car. — One  of  the  most  wonderful  recent  mechanical  achieve- 
ments is  the  successful  operation  of  a  car  which  runs  on  a  track  consisting 
of  only  one  rail.  By  a  clever  adaptation  of  the  gyroscopic  principle  of 
precession  of  two  wheels  having  opposite  rotation  (the  "Gyrostat"),  the 
car  is  balanced,  whether  in  motion  or  at  rest.  In  rounding  a  curve, 
the  "Gyrostat"  causes  the  car  to  "lean  in"  just  the  right  amount 
(Sec.  62). 

If  the  passengers  move  to  one  side  of  the  car,  that  side  of  the  car  rises, 
paradoxical  though  it  may  seem,  and  the  equilibrium  is  maintained.  As 
the  passengers  move  to  the  side,  a  "table  "  presses  on  the  axis  of  the  wheel , 
which  axis  is  transverse  to  the  car,  and  through  the  friction  developed 
by  the  rotation  of  the  axis,  against  the  table,  the  end  of  the  axis  is  caused 
to  creep  forward  (or  backward)  thus  developing  a  torque  about  a  vertical 
axis,  and  precession  about  an  axis  at  right  angles  to  both  of  these,  namely, 
an  axis  lengthwise  of  the  car.  This  precession  gives  rise  to  the  torque 
that  raises  higher  the  heavier  loaded  side  of  the  car.  For  an  extended 
discussion  of  the  gyroscope  and  numerous  illustrations  and  practical 
applications,  consult  Spinney's  Textbook  of  Physics,  or  Franklin  and 
MacNutt's  Mechanics  and  Heat. 

64.  Simple  Harmonic  Motion. — Simple  harmonic  motion 
(S.H.M.)  is  a  very  important  kind  of  motion  because  it  is  quite 
closely  approximated  by  many  vibrating  bodies.  Thus  if  a 
mass,  suspended  by  a  spiral  spring,  is  displaced  from  its  equilib- 
rium position  and  then  released,  it  will  vibrate  up  and  down  for 
some  time,  and  its  motion  will  be  simple  harmonic  motion. 
Other  examples  are  the  vibratory  motions  of  strings,  and  reeds 
in  musical  instruments,  the  vibratory  motion  of  the  air  (called 
sound)  which  is  produced  by  strings  or  other  vibrating  bodies, 
and  the  motion  of  the  simple  pendulum. 

The  vibrations  of  the  string  of  a  musical  instrument  consist, 
as  a  rule,  of  a  combination  of  vibrations  of  the  string  as  a  whole, 


UNIFORM  CIRCULAR  MOTION  83 

and  vibrations  of  certain  portions  or  segments.  Consequently 
the  motion  of  a  vibrating  string  is  usually  a  combination  of  several 
simple  harmonic  motions.  We  shall  here  restrict  ourselves  to 
the  study  of  the  simpler  case  of  uncombined  S.H.M. 

The  piston  of  a  steam  engine  executes  approximately  S.H.M.; 
while  in  the  motion  of  the  shadow  of  the  crank  pin  cast  upon  a 
level  floor  by  the  sun  when  over  head,  we  have  a  perfect  example 
of  S.H.M.  Observe  that  the  motion  of  the  crank  pin  itself  is 
not  S.H.M.,  but  uniform  circular  motion.  An  exact  notion  of 
what  S.H.M.  is,  and  a  simple  de- 
duction of  its  important  laws,  are 
most  readily  obtained  from  the  fol- 
lowing definition,  which,  it  will  be 
seen,  accords  with  the  statement 
just  made  with  regard  to  the  crank 
pin.  S.H.M.  is  the  projection  of 
uniform  circular  motion  upon  a  di- 
ameter of  the  circle  described  by  the 
moving  body. 

To  illustrate  the  meaning  of  the 
above  definition,  let  A  (Fig.  36)  be 
a  body  traveling  with  uniform  speed 
in  the  circular  path  as  shown.  Let 

DC  be  any  chosen  diameter,  say  a  horizontal  diameter.  From 
A  drop  a  perpendicular  on  DC.  The  foot  B  of  this  perpendicular 
is  the  "projection"  of  A.  Now  as  A  moves  farther  toward  D,  B 
moves  to  the  left  at  such  a  rate  as  always  to  keep  directly  below 
A.  As  A  moves  from  D  back  through  F  to  C,  B  constantly 
keeps  directly  above  A.  Under  these  conditions  the  motion  of 
B  is  S.H.M. 

In  the  position  shown  it  will  be  evident  that  B,  in  order  to 
keep  under  A,  need  not  move  so  fast  as  A.  When  A  reaches 
E,  however.  B  will  be  at  0  and  will  then  have  its  maximum  speed, 
which  will  be  equal  to  A's  speed.  As  A  and  hence  B  approach 
D,  the  speed  of  B  decreases  to  zero.  In  case  of  the  vibrating 
mass  supported  by  a  spring  (mentioned  at  the  beginning  of  the 
section)  it  is  evident  that  its  velocity  would  be  zero  at  both  ends 
of  its  vibration  and  a  maximum  at  the  middle,  just  as  we  have 
here  shown  to  be  the  case  with  B. 

66.  Acceleration  and  Force  of  Restitution  in  S.H.M.— If  the 
two  bodies  A  and  B  move  as  described  in  Sec.  64,  it  is  clear  that 


84  MECHANICS  AND  HEAT 

they  both  have  at  any  and  every  instant  the  same  horizontal 
velocities.     Thus,  in  the  position  shown  in  Fig.  36,  we  see  that 


B's  velocity  (horizontal)  must  be  equal  to  the  horizontal  com- 
ponent of  A's  velocity.  An  instant  later,  A's  horizontal  compo- 
nent of  velocity  will  have  increased,  and  since  B  always  keeps 
directly  below  (or  above)  A,  we  see  that  B's  velocity  must  have 
increased  by  the  same  amount.  In  other  words,  the  rate  of 
change  of  horizontal  velocity,  or  the  horizontal  acceleration,  is  the 
same  for  both  bodies. 

Similar  reasoning  shows  that  as  A  passes  from  E  to  D,  and 
consequently  B  passes  from  0  to  D,  the  leftward  velocity  de- 
creases at  the  same  rate  for  both  bodies.  As  A  passes  from  D 
to  F  and  then  from  F  to  C,  we  see  that  B  passes  from  D  to  0  with 
ever  increasing  velocity,  and  then  from  0  to  C  with  decreasing 
velocity.  To  summarize,  we  may  state  that  at  every  instant 
the  horizontal  components  of  A's  velocity  and  acceleration  are  equal, 
respectively,  to  the  actual  (also  horizontal}  velocity  and  acceleration 
of  B  at  that  same  instant. 

We  have  seen  that  whenever  B  moves  toward  0,  its  velocity 
increases,  while  as  it  moves  away  from  0,  its  velocity  decreases; 
i.e.,  its  acceleration  is  always  toward  0.  To  impart  to  B  such 
motion,  obviously  requires  an  accelerating  force  always  pulling 
B  toward  O.  We  shall  presently  show  that  this  force,  called  the 
Force  of  Restitution  Fr,  is  directly  proportional  to  the  distance 
that  B  is  from  0.  This  distance  is  called  the  Displacement  x. 

The  central  force  required  to  cause  A  to  follow  its  circular 
path  is 


and  the  horizontal  component  of  this,  or  —  Fh,  has  the  value 
shown  in  Fig.  36.  Note  that  if  the  vector  x,  directed  to  the 
right,  is  positive,  then  Fh,  when  directed  to  the  left,  is  negative. 
Now  Fh  is  the  accelerating  force  that  gives  A  its  horizontal 
acceleration,  while  Fr  is  the  accelerating  force  that  gives  B  its 
horizontal  acceleration;  but  these  two  horizontal  accelerations 
have  been  shown  to  be  always  equal.  Hence  if  A  and  B  are  of 
equal  mass  M,  it  follows  from  F  =  Ma  (Eq.  5)  that  Fr=Fh. 
From  similar  triangles  (Fig.  36)  we  have 

-Fh/Fc  =  x/r,  i.e.,  -Fh  or  -FT  =  XFC  =  4^n2Mx        (41) 


UNIFORM  CIRCULAR  MOTION  85 

Eq.  41  shows  that  the  force  of  restitution,  acting  upon  B  at  any 
instant,  is  proportional  to  the  displacement  of  B  at  that  same 
instant.  Accordingly  B's  accelerating  force,  and  hence  its 
acceleration  or  rate  of  change  of  velocity,  is  a  maximum  when  at 
C  or  D,  at  which  points  its  velocity  is  zero,  and  a  minimum  (in 
fact  zero)  at  0,  at  which  point  B  has  its  maximum  velocity. 
The  minus  sign  indicates  that  the  force  of  restitution  is  always 
oppositely  directed  to  the  displacement.  Thus  when  B  is  toward 
the  left  from  0,  x  is  negative,  but  Fh  is  then  positive. 

If,  then,  a  body  is  supported  by  a  spring  or  otherwise,  in  such 
a  manner  that  the  force  required  to  displace  it  varies  directly 
as  the  displacement,  we  know  at  once  that  the  body  will  execute 
S.H.M.  if  displaced  and  then  released.  Thus  it  can  easily  be 
shown,  either  mathematically  (Sec.  67)  or  experimentally,  that 
the  force  required  to  displace  a  pendulum  bob  is  proportional 
to  the  displacement,  provided  the  latter  is  small.  Hence  we 
know  that  when  the  bob  is  released  it  will  vibrate  to  and  fro  in 
S.H.M. 

66.  Period  in  S.H.M.—  Solving  Eq.  41  for  n  gives 


If  a  body  makes  n  vibrations  per  second,  its  period  of  vibration, 
or  the  time  P  required  for  one  complete  vibration  (a  swing  to 
and  fro),  is  1/n;  hence 


P  =  2Tr-^  (42) 

Eq.  42  gives  the  period  of  vibration  for  any  body  executing 
S.H.M.,  i.e.,  for  any  body  for  which  the  force  of  restitution  is 
proportional  to  the  displacement  x,  and  in  such  a  direction  as  to 
oppose  the  displacement.  In  this  equation,  M  is  the  mass  of 
the  vibrating  body  in  grams,  P  the  period  of  vibration  in  seconds, 
and  Fr  the  force  of  restitution  in  dynes,  when  the  displacement 
is  x  centimeters.  See  remark  on  units  below  Eq.  36,  Sec  58  and 
also  Sec.  32.  Since  x  and  Fr  always  differ  in  sign,  the  expression 
under  the  radical  sign  is  intrinsically  positive. 

If  a  heavy  mass  suspended  by  a  spiral  spring  requires  a  force 
of  1  kilogram  to  pull  it  downward,  say,  1  cm.  from  its  equilib- 
rium position,  it  will  require  a  force  of  2  kilograms  to  displace 
it  (either  downward  or  upward)  2  cm.  from  its  equilibrium  posi- 


86 


MECHANICS  AND  HEAT 


tion.  This  shows  that  the  force  of  restitution  is  proportional 
to  the  displacement;  hence  we  know  that  if  the  mass  is  pulled 
down  and  suddenly  released,  it  will  vibrate  up  and  down  and  exe- 
cute S.H.M.  Here  Fr  or 2  X 1000 X 980  =  1,960,000  dynes  when  x  = 
2  cm.  Suppose  that  the  mass  is  3  kilograms.  We  may  then 
find  its  period  of  vibration,  without  timing  it,  by  substituting 
these  values  in  Eq.  42.  Thus,  neglecting  the  mass  of  the  spring, 

=  0.346  sec. 


67.  The  Simple  Gravity  Pendulum. — The  following  discussion 
applies,  to  a  very  close  degree  of  approximation,  to  the  physical 
simple  pendulum  having  a  small  bob 
B  (Fig.  37)  suspended  by  a  light  cord 
or  wire.  The  length  L  of  the  pendu- 
lum is  the  distance  from  the  center 
of  the  bob  to  the  point  of  suspension. 
Consider  the  force  upon  the  bob  at 
some  particular  point  in  its  path.  Its 
weight,  W,  or  Mg  (Fig.  37),  may  be 
resolved  into  two  components,  7^1  in 
the  direction  of  the  suspending  wire, 
and  FT  in  the  direction  of  motion,  i.e,. 
toward  A.  For  small  values  of  6,  A 
approaches  0,  so  that  CA  may  be 
called  equal  to  CO,  i.e.,  equal  to  L, 
and  Fr  may  be  called  the  force  of 
restitution.  From  similar  triangles, 


FIG.  37. 


FT         -X  FT          -X 

W  =  CA  or  approx'  'Mg  =  ~L    or 


-Mgx 
L 


(43) 


Eq.  43  shows  that  the  force  of  restitution  Fr  is  proportional 
to  the  displacement,  and  oppositely  directed.  Hence  the  pen- 
dulum executes  S.H.M. ;  and  we  may  therefore  substitute  the 
value  of  Fr  from  Eq.  43  in  Eq.  42,  and  obtain  the  period  P  of 
the  pendulum, 


- 

2ir\ 
\  - 


-MX 

jrr- 
Mgx 


(44) 


UNIFORM  CIRCULAR  MOTION  87 

The  maximum  value  of  x,  i.e.,  the  distance  from  0  to  the  bob 
when  at  the  end  of  the  swing,  is  called  the  Amplitude  of  vibration. 
Since  x  and  M  cancel  out  in  Eq.  44,  the  period  of  a  pendulum  is 
seen  to  be  independent  of  either  its  mass  or  its  amplitude.  The 
latter  is  true  only  for  small  amplitudes.  If  x  is  large,  CA  and 
CO  are  not  approximately  equal,  as  is  assumed  in  the  above 
derivation.  A  pendulum  vibrates  somewhat  more  slowly  if 
the  amplitude  is  large  than  if  it  is  small,  since  CA  appreciably 
exceeds  L  when  6  is  large,  thus  making  Fr  smaller  than  given 
by  Eq.  43. 

68.  The  Torsion  Pendulum. — The  torsion  pendulum  usually 
consists  of  a  heavy  disc  suspended  from  its  center  by  a  steel 
wire,  and  hence  free  to  rotate  in  a  horizontal  plane.  When  the 
disc  is  rotated  from  its  equilibrium  position  through  an  angle 
6,  it  is  found  that  the  resisting  torque  is  proportional  to  the  angle 
0.  In  this  case,  the  torque  of  restitution,  or  the  returning  torque, 
is  (a)  proportional  to  the  displacement  angle  0,  and  (6)  opposes 
the  displacement.  These  are  the  two  conditions  for  S.H.M.  of 
rotation.  In  the  case  of  the  balance  wheel  of  a  watch,  the  torque 
of  restitution  due  to  the  hair  spring,  is  proportional  to  the  angle 
through  which  the  balance  wheel  is  rotated  from  its  equilibrium 
position.  Hence  the  balance  wheel  of  a  watch  executes  S.H.M. , 
and  therefore  its  period  is  independent  of  the  amplitude  of  its 
rotary  vibration. 

PROBLEMS 

1.  If  a  2-lb.  mass  is  whirled  around  240  times  per  minute  by  means  of  a 
cord  4  ft.  in  length  (a)  what  is  the  pull  on  the  cord?     (6)  What  is  the  radial 
acceleration  experienced  by  the  mass? 

2.  A  mass  of  1  kilogram  is  whirled  around  180  times  per  minute  by 
means  of  a  cord  1  meter  in  length.     What  is  the  pull  on  the  cord?     (a) 
in  dynes?     (6)  in  grams  force?     (c)  What  is  tbe  radial  acceleration? 

3.  How  many  times  as  large  does  the  central  force  become  when  the 
velocity  (Prob.  2)  is  doubled?     When  r,  the  length  of  the  cord,  is  doubled, 
the  number  of   revolutions  per   second,  and   also   the   mass,   remaining 
the  same? 

4.  An  emery  wheel  12  in.  in  diameter  makes  2400  R.P.M.     Find  the 
force  (in  poundals,  and  also  in  pounds),  acting  upon  each  pound  mass  of  tbe 
rim  of  the  wheel  tending  to  "burst"  it. 

6.  At  a  point  where  the  radius  of  curvature  r  (Fig.  34)  is  2000  ft.,  what 
is  the  "proper"  elevation  of  the  outer  rail  for  a  train  rounding  the  curve 
at  a  velocity  of  30  miles  per  hr.,  i.e.,  44  ft.  per  sec.?  Distance  between 
rails  is  4  ft.  8  in. 

6.  An  occupant  of  a  ferris  wheel  20  ft.  from  its  axis  observes  that  he 


88  MECHANICS  AND  HEAT 

apparently  has  no  weight  when  at  the  highest  point.  Find  his  linear 
velocity  and  radial  acceleration,  and  also  the  angular  velocity  of  the 
wheel. 

7.  Find  the  maximum  velocity  of  the  occupant  of  a  20-ft.  swing  if  the 
pull  he  exerts  upon  the  swing  at  the  instant  the  ropes  are  vertical  is  one- 
half  more  than  his  weight. 

8.  The    diameter    of    a   cream  separator   bowl    is    20    cm.     Find    its 
"efficiency"  when  making  4800  R.P.M. 

9.  A  4000-gm.  mass,  when  suspended  by  a  spring,  causes  the  spring  to 
elongate  2  cm.     What  will   be  the   period  of  vibration  of   the   mass  if 
set  vibrating  vertically?     Neglect  the  mass  of  the  spring. 

10.  A  sprinter  passing  a  turn  in  the  path,  where  the  radius  of  curvature 
is  60  ft.,  at  a  speed  of  10  yds.  per  sec.,  leans  in  from  the  vertical  by  an  angle 
e.     Find  tan  6. 

11.  What  is  the  period  of  a  pendulum  (in  Lat.  45°)  which  has  a  length 
of  20  cm.?     100  cm.? 

12.  What  is  the  length  of  a  pendulum  (Lat.  45°)   that  beats  seconds, 
i.e.,  whose  full  period  of  vibration  is  2  sec.? 

13.  A   pendulum   30   ft.    in    length   has   a  period   of   6.0655    sec.    at 
London.     What  is  the  value  of  g  there? 

14.  A  pendulum  whose  length  is  10  meters  makes  567.47  complete 
vibrations  per  hour  at  Paris.     Find  the  value  of  g  at  Paris. 


v 

v< 


CHAPTER  VI 
WORK,  ENERGY  AND  POWER 


69.  Work. — Work  is  defined  as  the  production  of  motion 
against  a  resisting  force.  The  work  done  by  a  force  in  moving  a 
body  is  measured  by  the  .product  of  the  force,  and  the  distance 
the  body  moves,  provided  the  motion  is  in  the  direct-ion  of  the 
force  (see  Sec.  71).  Hence  work  W  may  always  be  expressed  by 
the  equation 

W=Fd  (45) 

Thus  the  work  done  by  a  team  in  harrowing  an  acre  of  ground  is 
equal  to  the  product  of  the  average  force  required  to  pull  the 
harrow,  and  the  distance  the  harrow  moves.  To  harrow  two 
acres  would  require  twice  as  much  work,  because  the  distance 
involved  would  obviously  be  twice  as  great.  If  the  applied  force 
is  not  sufficient  to  move  the  body,  it  does  no  work  upon  the  body. 
Thus  if  a  man  pushes  upon  a  truck,  it  does  not  matter  how  hard 
he  pushes,  nor  how  long,  nor  how  tired  he  becomes;  he  does  no 
work  upon  the  truck  unless  it  moves  in  response  to  the  push. 

In  case  F  and  d  are  oppositely  directed,  i.e.,  in  case  the  body, 
due  to  previous  motion  or  any  other  cause,  moves  a  distance  d 
against  the  force,  then  work  is  said  to  be  done  by  the  body  against 
the  force.  Thus  if  a  stone  is  thrown  upward,  it  rises  a  certain 
height  because  of  its  initial  velocity,  and  in  rising  it  does  work 
(Fd)  against  the  force  of  gravity.  As  it  falls  back  the  force  of 
gravity  does  work  (Fd)  upon  the  stone  in  accelerating  it. 

From  the  above  discussion,  we  see  that  work  may  be  applied 
in  three  general  ways;  viz.,  (a)  to  move  a  body  against  friction, 
(6)  to  move  it  against  some  force  other  than  friction,  e.g.,  as  in 
lifting  a  body,  and  (c)  to  accelerate  a  body,  i.e.,  to  impart  velocity 
to  it.  Observe  that  in  all  three  cases  the  applied  force  does  work 
against  some  equal  opposing  force.  In  case  (a)  it  is  the  friction 
force  Ff,  in  case  (6)  the  weight  W  (or  a  component  of  the  weight), 
and  in  case  (c)  the  inertia  force  Fi,  against  which  the  applied 
force  does  work. 

89 


90  MECHANICS  AND  HEAT 

As  a  train  starts  upgrade  from  a  station  and  traverses  a  distance 
d,  the  pull  FI  upon  the  drawbar  of  the  locomotive  does  work  in 
each  of  these  three  ways.  Calling  the  average  total  friction 
force  on  the  train  Ff,  the  component  of  the  weight  of  the  train 
which  tends  to  make  it  run  down  grade  Fw  (see  Fig.  8,  Sec.  19), 
and  the  average  inertia  force  or  resistance  which  the  train  offers 
to  being  accelerated  Fi}  we  have 

Total  work  Fld=Ffd+Fwd+Fid  (46) 

If,  at  the  above  distance  d  from  the  station,  the  drawbar  of  the 
locomotive  becomes  uncoupled  from  the  train  while  going  full 
speed  up  grade,  and  if  the  train  comes  to  rest  after  going  a  distance 
d',  it  is  clear  that  the  driving  inertia  force  F'i  of  the  train  (Sec.  43) 
does  work  F'jd'  in  pushing  the  train  up  the  grade  against  F/  and 
Fw,  so  that  the  work 

Fi'd'=Ffd'-\-Fwdf  (47) 

Observe  that  d  and  d'  and  also  Ft  and  F'i  would,  in  general,  be 
quite  different  in  value,  while  the  values  of  F/  and  Fw  would  be 
practically  the  same  before  and  after  uncoupling;  hence  these 
same  symbols  are  retained  in  Eq.  47. 

70.  Units  of  Work. — Since  force  may  be  expressed  in  dynes, 
grams,  poundals,  pounds,  or  tons,  and  distance  in  centimeters, 
inches,  or  feet,  it  follows  that  work,  which  is  force  times  distance, 
may  be  expressed  in  dyne-centimeters  or  ergs,  gram-centimeters, 
foot-poundals,  foot-pounds,  foot-tons,  etc.  Thus,  if  a  locomotive 
maintains  a  1-ton  pull  on  the  drawbar  for  a  distance  of  one  mile, 
the  work  done  is  5280  ft.-tons,  or  10,560,000  ft.-lbs.  If  a  20-lb, 
mass  is  raised  a  vertical  distance  of  5  ft.,  the  work  done  against 
gravitational  attraction  is  100  ft.-lbs.  If  a  force  of  60  dynes 
moves  a  body  4  cm.,  it  does  240  ergs  (dyne-centimeters)  of  work. 
In  scientific  investigations,  the  erg  is  the  unit  usually  employed; 
in  engineering  calculations,  on  the  other  hand,  the  unit  is  the  foot- 
pound. The  work  done  by  an  electric  current  is  usually  com- 
puted in  joules.  One  joule  is  107  ergs. 

In  changing  from  one  work  unit  to  another,  it  must  be  observed 
that  work  contains  two  factors.  For  example,  let  it  be  required 
to  express  the  above  100  ft.-lbs.  of  work  in  terms  of  ergs.  This 
may  be  done  in  two  ways:  (1)  by  reducing  the  20-lb.  force  to 
dynes  and  the  5  ft.  to  centimeters,  and  then  multiplying  the  two 
results  together;  or  (2)  by  finding  the  number  of  ergs  in  a  foot- 


WORK,  ENERGY  AND  POWER  91 

pound  and  then  multiplying  this  number  by  100.  The  foot- 
pound is  larger  than  the  erg  for  two  reasons :  first,  1  foot  =  30.48 
centimeters,  and  second,  the  pound  being  approximately  453 
grams,  and  the  gram  force  being  980.6  dynes,  it  follows  that  the 
pound  force  =  445,000  dynes.  The  foot-pound  is  therefore 
30.48  X445,000  or  13,563,000  ergs.  Therefore  100  ft.-lbs.  of  work 
is  1,356,300,000  or  1.356X109  ergs. 

71.  Work  Done  if  the  Line  of  Motion  is  not  in  the  Direction  of 
the  Applied  Force. — In  Sec.  69  it  was  shown  that  work  =  Fd 
provided  F  and  d  have  either  the  same  direction  or  opposite  direc- 
tions, i.e.,  provided  the  angle  between  the  applied  force  and  the 
direction  of  motion  is  either  zero  or  180°.  If  this  angle  is  zero, 
then  work  is  done  by  the  force;  while  if  it  is  180°,  work  is  done 
against  the  force.  If  this  angle  is  90°,  no  work  is  done  either  by 
or  against  the  force.  Thus  if  a  team  is  pulling  a  wagon  westward, 
it  is  perfectly  obvious  that  a  man,  walking  along  side  the  wagon 
and  pushing  north  upon  it,  neither  helps  nor  hinders  the  team. 


FIG.  38. 

If  he  pushes  directly  forward,  the  above  angle  is  zero,  and  in 
traveling  a  distance  d  while  pushing  with  a  force  F  he  helps  the 
team  by  an  amount  of  work  Fd;  while  if  he  pulls  back  the  angle 
is  180°,  and  he  adds  Fd  to  the  work  the  team  must  do. 

If  he  pulls  slightly  to  the  south  of  west  with  a  force  F  (Fig.  38, 
top  view  of  wagon)  he  does  an  amount  of  work  which  is  less  than 
Fd.  Resolving  F  into  components  FI  and  Fz,  respectively 
parallel  and  perpendicular  to  the  line  of  motion,  we  see  that  F2 
simply  tends  to  overturn  the  wagon,  while  FI  is  fully  effective 
in  helping  the  team.  The  work  done  by  F  is  then  Fid,  but 
Fi=F  cos  e,  hence 

W=Fld=Fdcose  (48) 

As  6  approaches  90°,  cos  6,  and  hence  the  work  done,  approaches 
zero.  As  0  decreases,  i.e.,  as  the  man  pulls  more  nearly  west, 
cos  B  approaches  its  maximum  value,  unity  (when  0  =  zero),  and 
the  maximum  work  (Fd)  is  obtained.  Since  cos  180°=  —1,  we 


92  MECHANICS  AND  HEAT 

see  that  when  F  is  a  backward  pull  on  the  wagon,  then  W=  —  Fd. 
The  negative  sign  indicates  that  the  work  instead  of  being  done 
by  the  man,  is  added  work  done  by  the  team. 

72.  Work  Done  by  a  Torque. — If  the  force  F  (Fig.  39)  pushes 
the  crank  through  an  arc  AB,  the  work  done  is  force  times  dis- 
tance, or  W=FXAB.     But  by  definition 

,    arc     AB   ,  ... 

6  =  —  = ,  from  which  AB  =  rd; 

hence 

W=FXAB=Fr6 

But  since  torque  (T)  equals  force  times  radius, 

W=Td  (49) 

In  rotary  motion,  it  is  usually  more  convenient  to  compute 
work  by  means  of  Eq.  49  than  by  means  of  Eq.  45.     If  F  is  ex- 
pressed in  pounds  and  r  in  feet,  i.e.,  if 
the  torque  is  expressed  in   pound-feet, 
and  B  in  radians,  then  Td  gives  the  work 
done  in  foot-pounds.     Thus,  for  example, 
if  F  is  10  Ibs.,  r  is  2  ft.,  and  6  is  0.6  radi- 
ans, the  work  done  is  12  ft.-lbs.     If  T6 
is  expressed  in  C.G.S.  units  (dyne,  cen- 
p      on  timeter,  and  radian),  the  resulting  work 

is  given  in  ergs. 

73.  Energy — Potential  and  Kinetic. — The  energy  of  a  body 
may  be  denned  as  the  ability  of  the  body  to  do  work.     The 
potential  energy  of  a  body  is  its  ability  to  do  work  by  virtue  of  its 
position  or  condition.     The  Kinetic  Energy  of  a  body  is  its  ability 
to  do  work  by  virtue  of  its  motion. 

The  weights  of  a  clock  have  potential  energy  equal  to  the  work 
they  can  do  in  running  the  clock  while  they  descend.  Likewise 
the  main  spring  of  a  clock  or  watch,  when  wound,  has  potential 
energy  equal  to  the  work  it  can  do  as  it  unwinds.  The  water  in  a 
mill  pond  has  potential  energy.  Powder  and  coal  have  potential 
energy  before  ignition.  A  bended  bow  has  potential  energy. 
When  the  string  of  the  bow  is  released  and  the  arrow  is  in  flight, 
the  energy  then  possessed  by  the  arrow  is  kinetic.  Any  mass  in 
motion  has  kinetic  energy. 

The  immense  amount  of  kinetic  energy  possessed  by  a  rapidly 
moving  train  is  appreciated  only  in  case  of  a  derailment  or  a 


WORK,  ENERGY  AND  POWER  93 

collision.  The  kinetic  energy  of  a  cannon  projectile  enables  it  to 
do  work  in  piercing  heavy  steel  armor  plate  even  after  a  flight 
of  several  miles,  during  all  of  which  flight  it  does  work  against 
the  air  friction  upon  it.  The  work  done  upon  the  armor  plate 
of  the  target  ship  is  Fd;  in  which  F  is  the  enormous  force  (average 
value)  required  to  push  the  projectile  into  the  plate,  and  d  is 
the  distance  to  which  it  penetrates. 

74.  Transformation  and  Conservation  of  Energy. — Energy 
may  be  transformed  from  potential  to  kinetic  energy  and  vice 
versa,  or  from  kinetic  energy  into  heat,  or  by  a  suitable  heat 
engine,  e.g.,  the  steam  engine,  from  heat  to  kinetic  energy;  but 
whatever  transformation  it  experiences,  in  a  technical  sense,  none 
is  lost.  In  practice,  energy  is  lost,  as  far  as  useful  work  is  con- 
cerned, in  the  operation  of  all  machines,  through  friction  of  bear- 
ings, etc.  This  energy  spent  in  overcoming  friction  is  not  actually 
lost,  but  is  transformed  into  heat  energy  which  cannot  be  profit- 
ably reconverted  into  mechanical  energy.  In  all  cases  of  energy 
transformation,  the  energy  in  the  new  form  is  exactly  equal  in 
magnitude  to  the  energy  in  the  old  form.  This  fact,  that  energy 
can  neither  be  created  nor  destroyed,  is  referred  to  as  the  law  of 
the  Conservation  of  Energy.  This  law  is  of  great  importance,  as 
will  appear  from  time  to  time.  It  condemns  as  visionary  all 
perpetual  motion  machines  purporting  to  furnish  power  without 
having  a  source  of  energy.  Further,  since  it  is  impossible  to 
entirely  eliminate  friction,  a  perpetual  motion  machine  neither 
using  nor  furnishing  power  is  seen  to  be  an  impossibility.  The 
kinetic  energy  of  the  moving  parts  of  such  a  machine  would  soon 
be  transformed  by  friction  into  heat,  and  no  longer  exist  as  visible 
motion. 

The  conservation  of  energy  is  one  of  the  well-established  laws 
of  Physics,  and  is  frequently  used  as  a  basis  in  the  derivation  of 
equations,  and  in  various  lines  of  reasoning  such  as  just  given 
with  regard  to  perpetual  motion  machines.  From  the  conserva- 
tion of  energy,  we  see  that  to  give  a  body  a  certain  amount  of 
energy,  whether  potential  or  kinetic,  an  exactly  equivalent 
amount  of  work  must  be  done  on  the  body. 

We  may  now  state  in  slightly  different  form  than  that  used  in 
Sec.  69,  the  fact  that  the  work  done  upon  a  body  may  be  used  in 
three  ways:  (a)  to  move  the  body  against  friction;  (6)  to  give  the 
body  potential  energy;  and  (c)  to  give  the  body  kinetic  energy. 
These  three  amounts  of  work  done  by  the  locomotive  upon  the 


94  MECHANICS  AND  HEAT 

train  (Sec.  69)  are  represented  respectively  by  the  three  terms  of 
the  right-hand  member  of  Eq.  46.  Since  F^d  is  the  work  done  by 
the  locomotive  in  accelerating  the  train,  i.e.,  in  giving  it  its  veloc- 
ity and  hence  its  kinetic  energy,  it  follows,  from  the  conservation 
of  energy,  that  Ffd  is  the  kinetic  energy  of  the  train  just  as  it 
reaches  the  point  at  the  distance  d  from  the  station.  Hence, 
when  uncoupled  at  this  point,  this  kinetic  energy  does  an  equal 
amount  of  work  F'4'  in  forcing  the  train  on  up  the  grade.  Eq.  47 
shows  that  this  work  is  used  partly  (F/d')  in  driving  the  train  on 
against  friction,  and  partly  (Fwd')  in  giving  the  train  more 
potential  energy. 

It  should  be  emphasized,  that  in  the  transformation  of  kinetic 
energy  into  potential  energy,  and  vice  versa,  work  is  always  done. 
To  illustrate,  suppose  that  a  gun  of  length  d  feet  fires  a  projectile 
of  weight  W  pounds  vertically  to  a  height  h  feet.  Designating  by 
F  the  average  force  (in  pounds)  with  which  the  powder,  upon 
exploding,  pushes  upon  the  projectile,  and  ignoring  all  friction 
effects  (see  Dissipation  of  Energy,  Sec.  77)  we  have  Fd  foot-pounds 
for  the  work  done  in  giving  the  projectile  its  kinetic  energy,  and 
Wh  foot-pounds  (force  times  distance)  for  the  work  done  by  the 
kinetic  energy  of  the  projectile  in  raising  itself  to  the  height  h,  in 
which  position  its  potential  energy  (Ep)  is  a  maximum  and  has 
the  value  Wh  foot-pounds.  This  maximum  potential  energy  (Ep 
max.)  is  the  ability  the  projectile  has  to  do  work  by  virtue  of 
its  elevated  position,  and  it  does  this  work  Wh  (force  times 
distance)  while  descending,  in  causing  the  velocity  of  the  pro- 
jectile to  increase,  thereby  increasing  its  kinetic  energy.  This 
kinetic  energy  (Ek)  at  the  instant  of  striking  is  of  course  a 
maximum  (Ek  max.),  and,  by  the  conservation  of  energy,  it 
must  be  equal  to  the  work  Wh  done  by  gravitational  attraction 
in  giving  it  this  energy. 

To  summarize,  we  have,  in  accordance  with  the  conservation 
of  energv,  the  following  successive  energy  transformations: 
Fd  (work  done  by  powder)   =  E  max.  (at  muzzle)  =  work  Wh 
(done  while  rising)  =  Ep  max.  or  Wh  (at  highest  point)  =  work 
Wh  (done  while  descending)  =  Ek  max.  (at  striking). 

As  the  projectile  rises,  its  kinetic  energy  decreases,  while  its 
potential  energy  increases;  but,  from  the  conservation  of  energy, 
we  see  that  at  any  instant,  Ep-\-Ek  =  Wh  =  Ep  max.  Thus,  when 
the  projectile  is  at  a  height  \h,  it  is  evident  that  Ep  =  \Wh; 
hence,  at  that  same  instant,  it  must  be  that  Ek  =  %Wh.  If  h 


WORK,  ENERGY  AND  POWER  95 

were  10,000  ft.,  and  d,  10  ft.,  then  F  would  be  1000  times  the  weight 
of  the  projectile  (since  Fd=  WK).  Likewise,  if  a  1-ton  pile  driver 
falls  20  ft.  (19  ft.  before  striking)  and  drives  the  pile  1  ft.,  the 
average  force  on  the  pile  is,  barring  friction  effects,  20  tons.  The 
above  discussion  applies  to  the  similar  energy  transformations 
that  occur  in  the  operation  of  a  pile  driver,  and  in  the  vibration  of 
a  pendulum. 

76.  Value  of  Potential  and  Kinetic  Energy  in  Work  Units.— 
From  the  preceding  sections,  we  see  that  the  potential  energy,  or  \ 
the  kinetic  energy  possessed  by  a  body,  is  equal  to  the  work  (Fd)  \ 
required  to  give  it  that  energy.  Accordingly,  the  equation  express- 
ing the  potential  energy,  or  the  kinetic  energy  of  a  body  is  very 
simply  obtained  by  properly  expressing  this  work  (Fd).  In 
deriving  the  equation  for  potential  energy,  it  is  customary  to  take 
for  this  work,  the  work  (WK)  done  in  raising  a  mass  M  a  certain 
distance  against  gravitational  force;  while  for  the  kinetic  energy 
equation,  use  is  made  of  the  work  done  by  gravitational  force  on 
a  mass  M  in  falling  a  certain  distance.  This  is  done  for  two 
reasons:  first,  because  gravitational  potential  energy  is  the  kind 
of  potential  energy  with  which  we  have  to  deal  very  largely  in 
calculations,  while  the  kinetic  energy  of  falling  bodies  is  of  prime 
importance;  and  second,  because  of  the  fact  that  the  gravitational 
force  upon  a  body,  i.e.,  its  weight,  is  sensibly  constant  regardless 
of  change  of  height  or  velocity  of  the  body,  which  fact  very  much 
simplifies  the  derivations. 

The  Potential  energy  of  a  mass  M,  when  raised  a  height  h 
(Fig.  40),  is  equal  to  the  work  done  in  raising  it,  or  force  times 
distance.  Here  the  force  is  W  or  Mg,  and  the  distance  is  h,  so 
that 

Ep  =  Mgh  (50) 

Since  Mg  expresses  the  force  either  in  dynes  or  poundals  (Sec. 
32)  and  h  is  the  distance  either  in  centimeters  or  feet,  depending 
upon  which  system  is  used,  the  work,  and  hence  the  potential 
energy,  is  expressed  in  either  ergs  or  foot-poundals.  If  the  work 
is  wanted  in  foot-pounds,  the  weight  must  be  expressed  in 
pounds  and  the  distance  in  feet.  The  potential  energy  is  then 
given  by 

Ep  =  Mh  (50a) 

Note  that  a  mass  of  M  pounds  weighs  M  pounds,  not  Mg  pounds 
(Sec.  32). 


96  MECHANICS  AND  HEAT 

The  Kinetic  energy  of  a  moving  body  would  naturally  be 
expected  to  depend  upon  the  mass  of  the  body  and  upon  the  rapid- 
ity of  its  motion,  i.e.,  upon  its  velocity.  Suppose  that  the  body  of 
mass  M  (Fig.  40)  falls  the  distance  h.  Its  kinetic  energy  after 
having  fallen  that  height  must,  according  to  the  law  of  con- 
servation of  energy,  be  equal  to  the  work  done  upon  it  by  gravity 
while  falling,  or  force  Mg  times  the  distance  h.  Its  kinetic  energy 
is  then  Mgh,  which,  by  Eq.  50,  is  just  the  potential  energy  that 
it  has  lost  during  its  fall.  Substituting  for  h  its  value  for  falling 
bodies  given  in  Eq.  13,  Sec.  34,  namely,  h  =  %gt2,  gives 


Wvz  (51) 

If  the  English  system  is  used,  since  the  weight  or  force  is  expressed 
in  poundals,  the  result  obtained  by  substituting  the  mass  and 
velocity  of  the  moving  body,  in  Eq.  51, 
is  expressed  in  foot-poundals,  not  foot- 
pounds. If  M  is  the  mass  of  the  body  in 
grams,  then  Mg  is  the  force  in  dynes,  and  if 
h  is  expressed  in  centimeters,  Mgh,  and 
hence  the  kinetic  energy  %Mv2,  is  expressed 
in  dyne-centimeters  or  ergs. 

76.  Energy  of  a  Rotating  Body.  —  Any 
mass  particle  of  a  rotating  body,  e.g.,  a  fly- 
wheel, has  the  kinetic  energy  %mv2,  in 
which  m  is  the  mass  of  the  particle  and  v  its 
velocity.  Hence  the  kinetic  energy  of  the 


niE 


,    .      \~Ek^2-Mv*    entire  wheel  is  the  sum  of  all  the  quantities 
77M%  \mvi  for  each  and  all  of  its  mass  particles. 
FIG.  40.  Now  the  particles  near  the  rim  of  the  fly- 

wheel have  much  higher  velocities  and  hence 

much  greater  amounts  of  kinetic  energy  than  those  near  the  axis, 
so  that  the  actual  summation  of  the  kinetic  energy  for  all  particles 
cannot  be  effected  without  the  use  of  higher  mathematics.  We 
readily  see,  however,  that  two  wheels  of  equal  mass  M,  having 
equal  angular  velocity  co,  will  possess  different  amounts  of  kinetic 
energy  if  the  mass  is  mainly  in  the  rim  of  one  and  in  the  hub  of 
the  other.  Here,  as  in  so  many  other  cases,  a  very  simple  method 
of  deriving  the  expression  for  the  kinetic  energy  comes  from  the 
use  of  the  law  of  the  conservation  of  energy. 

From  this  law  we  know  that  the  kinetic  energy  Ek  of  the  fly- 
wheel, when  it  has  acquired  the  angular  velocity  co,  must  be 


WORK,  ENERGY  AND  POWER  97 

equal  to  the  work  T6  (Eq.  49,  Sec.  72)  done  by  the  applied  torque 
in  giving  it  this  kinetic  energy,  i.e.,  in  imparting  to  it  this  angular 
velocity  w.  Hence  Ek=T8,  which,  by  a  few  simple  substitu- 
tions, may  be  brought  into  a  form  involving  only  the  moment  of 
inertia  /  of  the  wheel,  and  its  angular  velocity  w.  From  Eq. 
30,  Sec.  54,  T  =  la, 

also          a=  y  (Sec.  51),  8=  at  (Eq.  27,  Sec.  51),  and  w  =  | 

Since  the  wheel  starts  from  rest  with  uniform  acceleration,  its 
average  angular  velocity  w  must  be  one-half  its  maximum 
angular  velocity  u>,  as  explained  in  Sec.  52.  Making  successively 
these  substitutions,  we  have 

Ek=T9  =  Iad  =  %Iuz  (52) 

If  we  use  C.G.S.  units  exclusively,  then  Td  (Eq.  52)  gives  the 
work  in  ergs  (Sec.  72)  required  to  produce  the  kinetic  energy 
l/co2,  which  energy  must  therefore  also  be  expressed  in  ergs.  I 
is  then,  of  course,  expressed  in  C.G.S.  units  or  gm.-cm.2  units 
(Sec.  55),  and  co  in  radians  per  second.  If  we  use  the  F.P.S. 
system  throughout,  then  Td  is  expressed  in  foot-poundals  (Sec. 
72),  £/a;2  in  foot-poundals,  o>  in  radians  per  second,  and  /  in 
lb.-ft.2  units  (Sec.  55). 

Let  us  now  apply  Eq.  52  to  find  the  kinetic  energy  of  the  1-ton 
flywheel  mentioned  in  Sec.  55,  when  co  =  20  radians  per  sec.,  i.e., 
when  the  flywheel  is  making  slightly  more  than  3  revolutions  per 
second.  The  moment  of  inertia  of  the  wheel  was  found  in 
Sec.  55  to  be  50,000  lb.-ft.2,  whence,  from  Eq.  52,  we  have  Ek  = 
|  50,000  X202=  10,000,000  foot-poundals  or  310,000  ft.-lbs. 
Dividing  this  energy  (310,000  ft.-lbs.)  by  550  (550  ft.-lbs.  per 
sec.  is  one  horse  power,  Sec.  82)  gives  562,  which  shows  that  the 
above  flywheel,  when  rotating  at  the  rate  of  20  radians  per  second, 
has  enough  kinetic  energy  to  furnish  1  horse  power  (H.P.)  in 
driving  the  machinery  for  562  seconds,  or  nearly  10  minutes, 
before  coming  to  rest. 

In  case  the  angular  velocity  of  a  flywheel,  connected  with  a 
gas  engine,  decreases  from  wi  just  after  an  explosion  stroke,  to 
co2  just  before  the  next  explosion  stroke,  then  the  energy  Ek 
which  it  gives  up  in  carrying  the  load  during  the  three  idle  strokes 
(Sec.  56)  is 

#*  =  $/wi2-iW,  or  -|/(Wl2-co22)  (53) 


98  MECHANICS  AND  HEAT 

If  the  wheel  makes  2  revolutions  per  sec.,  i.e.,  if  the  piston  makes 
4  strokes  per  sec.,  then  the  3  idle  strokes  will  last  3/4  second;  so 
that  if  the  engine  were  a  10-H.P.  engine,  the  work  W  which  the 
flywheel  would  have  to  do  in  this  3/4  second  would  be  550  X 
10X3/4  or  4125  ft.-lbs.  Evidently  this  work  W  equals  E  of  Eq. 
53,  or 

TF  =  I/Can2  -co22)  (54) 

Eq.  54  is  usually  employed  in  computing  the  proper  moment  of 
inertia  /  for  a  flywheel  working  under  certain  known  conditions. 
Thus,  if  we  know  the  horse  power  of  a  certain  gas  engine,  the  aver- 
age angular  velocity  co  of  its  flywheel  shaft,  and  the  permis- 
sible speed  variation  a>i  —  wa,  we  can  compute  both  W  (in  foot- 
poundals)  and  coi2—  co22;  then,  substituting  these  two  quantities 
in  Eq.  54,  we  may  solve  for  /.  Having  found  the  value  of  /  in 
lb.-ft.2  units,  we  may,  by  using  the  equation  I  =  Mr2  (Sec.  55), 
choose  a  certain  value  for  the  radius  r  of  the  flywheel  and  then 
solve  for  its  mass  M;  or  we  may  choose  a  value  for  M  and  then 
find  the  proper  value  for  r  in  order  to  make  the  wheel  meet  the 
above  requirements. 

If  a  small  car  and  a  hoop  of  equal  mass  are  permitted  to  run  down  the 
same  incline,  it  will  be  found  that  upon  reaching  the  bottom  of  the 
incline  the  velocity  of  the  hoop  will  be  about  7/10  that  of  the  car. 
Suppose  that  these  velocities  are  7  ft.  per  sec.  for  the  hoop  and  10  ft. 
per  sec.  for  the  car.  The  potential  energy  at  the  top  of  the  incline  was 
the  same  for  both  bodies,  hence  the  kinetic  energy  upon  reaching  the 
bottom  must  be  the  same  for  both  (conservation  of  energy).  The  hoop 
has  kinetic  energy  of  both  translation  and  rotation,  while  the  car, 
neglecting  the  slight  rotational  energy  of  its  wheels,  has  only  energy  of 
translation.  Consequently  we  have 


in  which  the  left  member  is  the  energy  of  the  hoop,  and  the  right  member 
that  of  the  car.  Solving,  we  find  that  half  of  the  energy  of  the  hoop  is 
rotational  energy,  that  is,  experiment  shows  that  %Iw*  =  %Mv2  for  the 
hoop. 

Mathematical  Proof.  —  Since  I  =  Mr2  (the  mass  M  of  the  hoop  considered 
to  be  all  in  its  "rim"  of  radius  r  (see  below,  Eq.  32,  Sec.  55),  and 
since  v  =  ru  (Eq.  29,  Sec.  52),  we  have 


which  was  to  be  proved. 


WORK,  ENERGY  AND  POWER  99 

A  sphere,  or  a  wheel  with  a  massive  hub,  would  travel  more  nearly  as 
fast  as  the  car,  because  in  such  case  the  mass  would  not  be  all  concen- 
trated in  the  "rim,"  and  consequently  the  moment  of  inertia,  and 
therefore  the  rotational  energy,  would  be  less  than  for  the  hoop. 

77.  Dissipation  of  Energy. — The  fact  that  the  energy  of  a  body, 
whether  potential  or  kinetic,  always  tends  to  disappear  as  such, 
is  a  matter  of  common  observation,  and  is  referred  to  as  the  prin- 
ciple of  the  Dissipation  of  Energy.     Thus  a  body,  for  example  a 
stone,  in  an  elevated  position  has  potential  energy.     If  released, 
the  stone  falls,  and  at  the  instant  of  striking  the  ground  its 
energy  is  kinetic.     An  instant  later  the  stone  lies  motionless  upon 
the  ground,  both  its  potential  energy  and  kinetic  energy  having 
disappeared. 

The  results  of  many  carefully  performed  experiments  lead  to 
the  conviction  that  in  the  above  case  no  energy  has  been  lost  (see 
conservation  of  energy) ;  but  that,  due  to  air  friction  while  falling, 
and  friction  against  the  ground  as  it  strikes,  the  stone  has  slightly 
warmed  itself,  the  air,  and  the  ground;  and  that  the  amount  of 
heat  energy  so  developed  is  exactly  equal  to  the  original  potential 
energy  of  the  stone.  This  example  illustrates  the  general  trend 
of  energy  change  throughout  nature;  viz.,  the  potential  energy  of 
a  body  tends  to  change  to  kinetic  energy.,  and  its  kinetic  energy  tends 
to  change  into  heat  energy.  The  relation  between  heat  and  other 
forms  of  energy  will  be  further  considered  in  the  study  of  heat,  but 
it  might  here  be  mentioned  that  778  ft.-lbs.  of  work  used  in 
stirring  1  pound  of  water  will  warm  it  1°  F.  Attention  is  also 
called  to  the  fact  that  the  hands  may  be  warmed  by  rubbing  them 
together,  and  that  primitive  man  lighted  his  fires  by  vigorously 
rubbing  one  piece  of  wood  against  another. 

A  vibrating  pendulum,  a  rotating  flywheel,  or  a  moving  train 
soon  loses  its  motion  if  no  power  is  applied.  These  are  good 
examples  of  the  dissipation  of  energy.  In  all  such  cases,  the 
potential  energy  or  the  kinetic  energy  of  the  body  is  transformed 
into  heat  through  the  work  done  by  the  body  against  friction. 

78.  Sliding  Friction. — If  one  body  is  forced  to  slide  upon 
another,  the  rubbing  together  of  the  two  surfaces  gives  rise  to  a 
resisting  force  which  always  opposes  the  motion  and  is  called 
friction.     It  may  also  be  called  the  force  of  friction.     Either  sur- 
face may  be  that  of  a  solid,  a  liquid,  or  a  gas.     Thus  in  drawing  a 
sled  on  a  cement  walk,  the  friction  is  between  two  solids,  steel  and 
cement,    In  the  passage  of  a.  boat  through  water,  the  friction  is  be- 


100  MECHANICS  AND  HEAT 

tween  a  solid  and  a  liquid,  i.e.,  between  the  sides  and  bottom  of 
the  boat  and  the  water.  In  the  case  of  the  aeroplane,  there  is  fric- 
tion between  the  canvas  planes  or  wings  and  the  air  through  which 
they  glide.  If  the  wind  in  the  higher  regions  of  the  atmosphere 
has  either  a  different  velocity  or  a  different  direction  than  the 
surface  wind,  there  will  be  friction  between  them.  In  all  cases, 
the  work  (Fd)  done  against  friction  is  the  product  of  the  frictional 
force  and  the  distance  of  sliding,  and  is  transformed  into  heat 
energy  (Sec.  77).  Bending  a  piece  of  wire  back  and  forth  rapidly, 
heats  it  because  of  the  Internal  Friction  between  its  molecules, 
which  are  thereby  forced  to  slide  past  each  other.  Internal 
friction  in  liquids  causes  them  to  become  heated  when  stirred, 
and  also  gives  rise  to  viscosity.  The  greater  viscosity  or  molecu- 
lar friction  of  syrups  makes  them  flow  much  more  slowly  than 
water. 

A  smooth  board  or  iron  plate  appears  rough  under  the  micro- 
scope due  to  innumerable  slight  irregularities.  The  cause  of 
friction  is  the  fitting  together  or  interlocking  of  these  irregularities 
of  one  surface  with  those  of  the  other  over  which  it  slides.  It  is 
easily  observed  that  it  takes  a  greater  force  to  start  the  sliding  of 
a  body  than  to  maintain  it.  The  former  force  must  overcome  the 
backward  drag  of  Static  Friction;  the  latter,  that  of  Kinetic 
Friction.  The  greater  value  of  static  friction  is  probably  due  to 
the  better  interlocking  of  the  irregularities  of  the  two  surfaces 
when  at  rest  than  when  in  motion  relatively  to  each  other.  This 
view  is  supported  by  the  fact  that  when  the  velocity  of  sliding  is 
very  small  the  kinetic  friction  differs  very  little  from  the  static. 

The  so-called  "Laws  of  Friction"  are:  (a)  the  friction  is  directly 
proportional  to  the  force  pressing  the  surfaces  together;  (6)  it  is 
independent  of  the  area  of  the  surfaces  in  contact;  and  (c)  it  is 
independent  of  the  velocity  of  sliding.  These  laws  are  approxi- 
mately true  between  wide  limits.  Thus  the  force  required  to 
draw  a  sled  will  be  approximately  doubled  by  doubling  the  load, 
will  be  very  little  affected  by  change  in  the  length  of  runner 
(within  reasonable  limits),  and  will  remain  about  the  same  though 
the  velocity  is  varied  from  1  mile  per  hour  or  less,  to  several  times 
that  value. 

To  reduce  the  waste  of  power  and  also  the  wearing  of  ma- 
chinery due  to  friction,  lubricating  oils  are  used.  The  film  of 
oil  between  the  two  rubbing  surfaces  prevents  their  coming  into 
such  intimate  contact,  and  thus  prevents,  in  a  large  measure,  the 


WORK,  ENERGY  AND  POWER  101 

interlocking  of  the  above-mentioned  irregularities.  During  the 
motion,  the  particles  of  oil  in  this  film  glide  over  each  other  with 
very  little  friction,  and  the  total  friction  is  thus  reduced  by  sub- 
stituting, in  part,  liquid  friction  for  sliding  friction.  The  resist- 
ance which  a  shaft  bearing  offers  to  the  rotation  of  the  shaft,  is 
evidently  sliding  friction,  and  is  therefore  reduced  by  proper  oiling. 

In  general,  friction  is  greater  between  two  surfaces  of  the  same 
material  than  it  is  between  those  of  different  materials.  Thus 
bearings  for  steel  shafts  are  sometimes  made  of  brass,  and  fre- 
quently of  babbitt,  to  reduce  friction.  Babbitt  metal  is  an  alloy 
of  tin  with  copper  and  antimony,  as  a  rule.  Sometimes  lead  is 
added.  On  the  other  hand,  iron  brake  shoes  are  used  on  iron 
wheels  to  obtain  a  large  amount  of  friction,  and  pulleys  are 
faced  with  leather  to  prevent  belt  slippage. 

The  wasteful  effects  of  friction  are  usually  apparent,  but  the 
beneficial  effects  are  probably  not  so  generally  appreciated.  If 
it  were  not  for  friction,  it  would  be  impossible  to  transmit  power 
by  means  of  belts,  or  to  walk  upon  a  smooth  surface.  Further- 
more, all  machinery  and  all  structures  which  are  held  together 
by  nails,  screws,  or  by  bolts  (unless  riveted),  would  fall  to  pieces 
instantly  if  all  friction  were  eliminated. 

79.  Coefficient  of  Friction.— The  Coefficient  of  Kinetic  Friction 
is  defined  as  the  ratio  of  the  force  required  to  move  a  body  slowly 
and  with  uniform  velocity  along  a  plane,  to  the  force  that  presses 
it  against  the  plane.  Thus,  if  a  force  of  30  Ibs.  applied  in  a  hori- 
zontal direction  is  just  sufficient  to  move  a  body  of  mass  100  Ibs. 
slowly  and  with  uniform  velocity  over  a  level  surface,  then  the 
coefficient  of  kinetic  friction  of  that  particular  body  upon  that 
particular  surface  is  30/100  or  0.3. 

A  very  simple  piece  of  apparatus  for  finding  the  coefficient  of 
friction  is  shown  in  Fig.  41.  B  is  a  board,  say  of  oak,  which 
may  be  inclined  at  such  an  angle  that  the  block  C,  say  of  walnut, 
will  slide  slowly  down  the  plane  due  to  its  weight.  Let  this  angle 
be  6.  Resolving  W,  the  weight  of  the  block  C,  into  two  compo^. 
nerits,  one  component  Fi  urging  it  along  the  plane,  and  the  other 
F2  pressing  it  against  the  plane,  we  have  by  definition  F\/Fz  as 
the  coefficient  of  friction.  Fi/Fz,  however,  is  also  tan  6,  hence 
for  this  type  of  apparatus  the 

Coeff.  of  f riction  =Fi/F»  =  tan  6. 
From  the  figure  it  is  seen  that  h/d  is  also  tan  6;  so  that  if  in  this 


102 


MECHANICS  AND  HEAT 


particular  case  h/d  =  1/3,  the  coefficient  of  friction  for  walnut  on 
oak  is  0.33  for  the  particular  specimens  tested. 

The  coefficient  of  friction  of  metal  on  metal  is,  as  a  rule,  some- 
what greater  than  0.2  for  smooth,  dry  surfaces.  Oiling  may 
reduce  this  to  as  low  as  0.04. 

If  the  coefficient  of  friction  between  the  locomotive  drivers  and 
the  rail  is  0.2,  then  the  maximum  pull,  or  "tractive  effort,"  which 
the  locomotive  can  exert  upon  the  drawbar,  is  about  0.2  of  the 
weight  carried  by  the  drivers.  Any  attempt  to  exceed  this, 
results  in  the  familiar  spinning  of  the  drivers.  For  the  same 
reason,  the  maximum  resistance  to  the  motion  of  a  car  that  can 
be  obtained  by  setting  the  brakes,  is  about  0.2  of  the  weight  of 
the  car.  Any  attempt  to  exceed  this  force  results  in  sliding,  with 
the  production  of  the  so-called  "flat"  wheel. 

The  Coefficient  of  Static  Friction  is  defined  as  the  ratio  between 
the  force  required  to  start  a  body  to  slide,  and  the  force  pressing  it 


FIG.  41. 

against  the  plane.  Since  it  requires  a  greater  force  to  start 
sliding  than  to  maintain  it,  the  coefficient  of  static  friction  is 
larger  than  the  coefficient  of  kinetic  friction  for  the  same  materials. 
The  probable  reason  for  this  difference  is  the  better  interlocking 
of  the  surfaces  in  the  case  of  static  friction  (Sec.  78). 

80.  Rolling  Friction. — It  is  a  matter  of  common  knowledge 
that  to  draw  a  1000-lb.  sled,  having  steel  runners,  along  a  steel 
track  would  require  a  much  greater  force  than  to  draw  a  1000-lb. 
truck,  having  steel  wheels,  along  the  same  track.  In  the  former 
case  sliding  friction  must  be  overcome;  in  the  latter  case,  rolling 
friction.  The  fact  that  rolling  friction  is  so  much  smaller  than 
sliding  friction  has  led  to  the  quite  common  use  of  ball  bearings 
in  machinery.  Thus  the  wheel  of  a  bicycle  or  of  an  automobile 
supports  the  axle  by  means  of  a  train  of  very  hard  steel  balls  of 
uniform  size,  which  are  free  to  roll  round  and  round  in  a  groove 


WORK,  ENERGY  AND  POWER  103 

on  the  inside  of  the  hub  as  the  wheel  turns.  The  axle  rests  with 
a  similar  groove  upon  these  balls  and  is  thereby  prevented  from 
direct  rubbing  (sliding  friction)  against  the  hub.  Recent  Ameri- 
can practice  favors  rollers  instead  of  balls  for  automobile  "anti- 
friction" bearings.  By  means  of  ball  bearings,  the  coefficient 
of  friction,  so-called,  may  be  reduced  to  about  1/2  per  cent. 

In  drawing  the  above  truck  on  the  steel  track,  the  resistance 
encountered  is  due  to  the  fact  that  the  steel  wheel  makes  a. slight 
depression  in  the  rail,  and  is  itself  slightly  flattened  by  the  weight. 
Since  the  material  in  the  rail  is  not  perfectly  elastic,  the  minute 
"  hill "  in  front  of  the  wheel  is  larger  than  the  one  behind  it.  The 
wheel  is  constantly  crushing  down  a  small  "hill"  A  in  front  of  it 
(shown  greatly  exaggerated  in  Fig.  42),  and  the  energy  required  to 
do  this  is  always  greater  than  the 
energy  applied  by  the  small  "hill" 
B  that  is  springing  up  behind  it. 

Since  the  thrust  a,  due  to  "hill" 
A,  is  greater  than  the  thrust  6,  due 
to  B,  the  general  upward  thrust  of 
the  rail  against  the  wheel  inclines 
very  slightly  backward  from  the 
vertical  as  shown.  If  the  weight 
W,  and  the  pull  F  necessary  to  J?IQ  42 

make    the    wheel    roll,    are    both 

known,  the  thrust  T  can  easily  be  determined  For,  since 
the  wheel  is  in  equilibrium,  the  three  forces  W}  F,  and  T, 
acting  upon  it  must  form  a  closed  vector  triangle.  If,  then,  W 
and  F  are  drawn  to  scale  as  shown,  the  closing  side  T  of  the  tri- 
angle represents  the  required  thrust.  In  the  case  of  car  wheels 
on  a  steel  track,  F  is  about  1  per  cent,  of  W,  so  that  the  angle  0 
is  really  much  smaller  than  shown.  In  the  case  of  a  rubber  wheel 
rolling  on  a  steel  rail,  the  depression  of  the  rail  would  be  prac- 
tically zero;  but  in  this  case  there  would  be  a  "bump"  on  the 
wheel  itself  just  in  front  of  the  flat  portion,  which  would  have  to 
be  crushed  down  as  the  wheel  advanced.  To  be  sure,  the  spring- 
ing out  of  the  rubber  "bump"  just  behind  the  flat  portion  would 
help  the  wheel  forward  just  as  the  rising  of  the  minute  hill  on 
the  rail  just  behind  the  wheel  would  help  it  forward  (in  case  the 
rail  is  depressed).  Since  rubber  is  not  perfectly  elastic,  the 
energy  required  to  crush  the  one  "bump"  is  greater  than  that 
obtained  from  the  other  formed  by  the  rubber  in  springing  out 


104  MECHANICS  AND  HEAT 

again  behind  the  wheel.  The  difference  between  these  two 
amounts  of  energy  is  the  energy  used  in  overcoming  rolling 
friction. 

If  the  wheel  and  the  rail  are  made  of  very  hard  steel,  friction  is 
reduced,  because  the  depression  made  is  less;  but  the  danger  of 
accidents  from  the  breaking  of  brittle  rails  is  increased.  In  the 
case  of  a  wagon  being  drawn  on  the  level  along  a  soft  spongy 
road,  the  conditions  are  the  same  as  those  just  discussed,  except 
that  the  "hill"  is  more  marked  in  front  of  the  wheel,  and  the  ris- 
ing of  the  hill  behind  the  wheel  is  extremely  sluggish  indeed.  For 
this  reason,  rolling  friction  is  a  vastly  greater  factor  in  wagon 
traffic  than  in  railway  traffic,  and  for  the  same  reason,  slight 
grades,  which  would  be  prohibitive  in  railway  traffic,  are  in  wagon 
traffic  of  small  importance  as  compared  with  the  character  of  the 
road  bed. 

The  friction  upon  the  axle  of  the  car  is  simply  sliding  friction, 
but  the  amount  of  energy  required  to  overcome  it  is  very  much 
less  than  if  the  sliding  were  directly  upon  the  rail  itself,  by 
means  of  a  shoe,  for  example.  If  the  diameter  of  the  axle  is  1/10 
that  of  the  wheel,  the  distance  of  sliding  between  the  axle  and  the 
hub  is  clearly  1/10  the  distance  traversed  by  the  car.  Hence 
we  see  that  the  work  required  to  overcome  this  friction  is  only 
1/10  as  much  as  it  would  be  if  the  sliding  were  directly  upon  the 
rail,  and  if  oil  were  sufficiently  cheap  to  maintain  as  good  lubrica- 
tion between  rail  and  shoe  as  is  maintained  on  axles. 

81.  Power. — Power  is  denned  as  the  rate  of  doing  work;  con- 
sequently average  power  is  the  work  done  divided  by  the  time 
required  to  do  the  work,  or,  proper  units  being  chosen, 

P  =  W/t  (55) 

If  the  work  done  in  t  seconds  is  divided  by  t,  the  result  is  the  work 
done  in  one  second.  Hence  power  is  numerically  the  work  done 
per  unit  time  (usually  the  second).  Thus  if  a  man  lifts  a  50-lb. 
weight  to  a  height  of  6  ft.  in  2  sec.,  he  does  300  ft.-lbs.  of  work. 
Dividing  this  amount  of  work  by  the  time  required  to  do  it 
gives  the  power  or  150  ft.-lbs.  per  sec.  Also  multiplying  the 
force,  50  Ibs.,  by  the  velocity,  3  ft.  per  sec.,  gives  likewise  150 
ft.-lbs.  per  sec.  For,  since  distance  d  =  vt,  we  have 


WORK,  ENERGY  AND  POWER  105 

or  power  is  equal  to  the  force  applied  multiplied  by  the  velocity  of 
motion  of  the  body  to  which  it  is  applied,  provided  the  motion 
is  in  the  direction  of  the  force.  Thus,  multiplying  the  pull  on  the 
drawbar  of  a  locomotive  in  pounds,  by  the  velocity  of  the  loco- 
motive in  feet  per  second,  gives  at  any  instant  the  power  de- 
veloped by  the  locomotive  in  foot-pounds  per  second. 

82.  Units  of  Power. — Since  power  is  the  rate  of  doing  work,  it 
must  be  expressed  in  terms  of  work  units  and  time  units,  e.g., 
ergs  per  second,  foot-pounds  per  second,  foot-pounds  per  minute, 
etc.  The  horse  power  (H.P.)  is  one  of  the  large  power  units  in 
common  use. 

1  H.P.  =  550  ft.-lbs.  per  sec.  =33,000  ft.-lbs.  per  min. 

Since  the  pound  force  or  pound  weight  increases  with  g,  it 
follows  that  the  horse  power  becomes  a  larger  unit  with  increase  of 
g.  Strictly,  the  standard  H.P.  is  550  ft.-lbs.  per  sec.  at  latitude 
45°  (g  =  980.6) .  At  latitude  60°,  e.g.,  in  central  Sweden  and  Nor- 
way, g  is  about  1/10  per  cent,  greater  than  at  latitude  45°,  so 
that  the  H.P.  there  used  is  about  1/10  per  cent,  larger  unit  than 
the  standard  H.P.,  unless  corrected.  Such  correction  is  not  made 
in  practice,  because  it  is  small  in  comparison  with  the  fluctuations 
in  power  that  occur  during  a  test  of  an  engine  or  motor. 

If  a  140-lb.  man  ascends  a  stairway  at  the  rate  of  4  ft.  (verti- 
cally) per  sec.,  the  work  done  per  second,  i.e.,  the  power  he  de- 
velops, is  560  ft.-lbs.  per  sec.,  or  slightly  more  than  1  H.P. 

If  a  span  of  horses,  pulling  a  loaded  wagon  weighing  2  tons  up 
a  hill  rising  1  ft.  in  10,  travels  at  the  rate  of  5  ft.  per  sec.,  then, 
since  the  load  rises  1/2  ft.  per  sec.,  the  power  developed  by  the 
two  horses  in  working  against  gravity  alone  is, 

4000X0.5  ft.-lbs.  per  sec.,  or  3.63  H.P. 

Considering  also  the  work  done  against  friction,  it  will  be  seen 
that  each  horse  would  probably  have  to  develop  more  than  2  H.P. 
The  above  unit  (550  ft.-lbs.  per  sec.)  expresses  the  power  which 
a  horse  can  develop  for  long  periods  of  time,  e.g.,  for  a  day.  It 
is^a  rather  high  value  for  the  average  horse.  On  the  other  hand, 
for  very  short  periods  (1/2  min.  or  so),  a  horse  may  develop  6  or 
8  H.P.  This  accounts  in  part  for  the  fact  that  a  30-H.P.  auto- 
mobile, stalled  in  the  sand,  may  readily  be  drawn  by  a  4-horse 
team.  It  may  be  mentioned  in  passing  that  the  French  H.P. 
of  75  kilogram-meters  per  sec.  is  541  ft.-lbs.  per  sec. 


106  MECHANICS  AND  HEAT 

Other  units  of  power  are  the  watt  (one  joule  per  sec.),  and  the 
kilowatt  (1000  watts).  These  units  are  used  extensively  in 
expressing  electrical  power.  The  H.P.  equals  approximately  746 
watts,  or  in  round  numbers,  3/4  kilowatt. 

From  Eq.  55  we  see  that  work  equals  power  times  time.  A 
span  of  horses  working  at  normal  rate  for  ten  hours  does  20  H.P.- 
hours  of  work.  A  good  steam  engine  will  do  1  H.P.-hour  of 
work  for  every  1.5  Ibs.  of  coal  burned.  If  the  lighting  of  a  certain 
building  requires  2  kilowatts  (K.W.)  then  the  energy  used  in  five 
hours  is  10  K.W.-hours.  This  energy  is  recorded  by  the  watt- 
hour  meter,  commonly  called  a  recording  wattmeter,  and  costs 
usually  about  ten  cents  per  K.W.-hour.  A  32-candle-power 
"carbon"  lamp  (i.e.,  a  lamp  whose  filament  is  made  of  carbon) 
requires  about  100  watts,  while  a  "tungsten"  lamp  having  the 
same  candle  power  requires  only  about  40  watts.  Observe 
in  this  connection  that  it  is  not  power  that  is  bought  or  sold,  but 
energy,  which  is  the  product  of  the  power  and  the  time. 

83.  Prony  Brake. — Various  devices  have  been  used  to  test  the 
power  of  steam  engines  and  motors.  With  some  of  them  the  test 
may  be  made  while  the  engine  is  doing  its  regular  work,  while 
others  require  that  the  regular  work  cease  during  the  test.  The 
Prony  Brake,  in  fact  all  brakes,  are  of  the  latter  class,  and  are 
known  as  absorption  dynamometers.  The  former  devices  are 
termed  transmission  dynamometers. 

Since  W  =Td  (Eq.  49),  and  $=ut, 

P  =  W/t=T  ut/t  =  Tu  (57) 

Hence  to  find  the  power  of  a  motor,  for  example,  it  is  merely  nec- 
essary to  find  what  torque  it  exerts,  and  then  multiply  this  by  its 
angular  velocity  co,  or  2-irn,  in  which  n  is  the  number  of  revolu- 
tions per  second  as  determined  by  a  speed  indicator  held  against 
the  end  of  the  motor  shaft.  A  strap  pressed  against  the  pulley  of 
the  motor  shaft  would  be  pulled  in  the  direction  of  rotation  with  a 
certain  force  F.  If  r  is  the  radius  of  the  pulley,  then  Fr  gives 
the  torque  of  Eq.  57.  Multiplying  this  torque  by  w,  as  above 
found,  would  give  the  power  of  the  motor  in  foot-pounds  per 
second,  provided  n  is  given  in  revolutions  per  second,  F  in  pounds, 
and  r  in  feet.  Dividing  this  result  by  550  would  then  give  the 
power  of  the  motor  in  H.P.  If  n  were  given  in  revolutions  per 
minute  (R.P.M.),  it  would  be  necessary  to  divide  by  33,000  in- 
stead of  by  550. 


WORK,  ENERGY  AND  POWER 


107 


A  simple  form  of  the  Prony  Brake,  suitable  for  testing  small 
motors  or  engines,  is  shown  in  Fig.  43.  The  pulley  A  of  the  motor 
shaft  is  clamped  between  two  pieces  of  wood,  B  and  C,  as  shown. 
The  end  D  of  C  is  attached  to  a  spring  balance  E.  As  the  pulley 
turns,  it  tends  to  rotate  the  brake  with  it,  but  is  prevented  by 
the  upward  pull  F  exerted  by  E  on  D.  The  force,  say  FI,  re- 
quired to  make  the  surface  of  the  pulley  slide  past  the  wood, 
times  the  radius  r\  of  the  pulley,  gives  the  driving  torque  F\TI 
tending  to  rotate  the  brake  in  a  clockwise  direction.  Since  the 
brake  does  not  rotate,  we  see  that  the  opposing  torque,  that  is, 
the  above  pull  F  times  its  lever  arm  r,  or  Fr,  must  equal  the 
torque  Ftfi.  Accordingly  the  former  torque  (Fr),  which  is  easily 
found,  may  be  used  in  Eq.  57. 


Fia.  43. 

If  B  and  C  are  lightly  clamped  together,  this  torque  will  be 
very  small,  making  the  power  small  (Eq.  57);  while  if  clamped 
too  tightly,  the  motor  may  be  so  greatly  slowed  down  that  the 
power  is  again  too  small.  The  proper  way  to  make  the  test  is 
to  gradually  tighten  the  clamp  until  the  electrical  instruments 
show  that  the  motor  is  using  its  rated  amount  of  electrical  power, 
and  then  take  simultaneous  readings  of  E  and  the  speed  indicator. 
From  these  readings  the  H.P.  of  the  motor  is  found  as  above  out- 
lined. Likewise  in  testing  a  steam  engine,  the  clamp  should 
be  tightened  until  both  the  speed  and  the  steam  consumption  are 
normal. 

In  testing  large  engines  or  motors  with  the  Prony  Brake,  D 
rests  on  a  platform  scale,  and  pulley  A,  in  some  cases,  has  a  rim 
projecting  inward  which  enables  it  to  hold  water  when  revolving, 


108  MECHANICS  AND  HEAT 

due  to  the  centrifugal  force  thereby  developed.  Water  applied 
in  this  or  some  other  way  prevents  undue  heating.  The  clamp 
also  differs  slightly  from  that  shown. 

A  convenient  form  of  brake  for  testing  small  motors  is  the 
Strap  Brake.  A  leather  strap  attached  to  one  spring  balance  is 
passed  down  around  the  motor  pulley  and  then  up  and  attached 
to  another  spring  balance.  Evidently  when  the  motor  is  running, 
the  two  spring  balances  will  register  different  forces.  The 
difference  between  these  two  forces  multiplied  by  the  radius  of  the 
pulley,  is  the  opposing  torque.  But  this  torque  is  equal  to  the 
driving  torque.  This  driving  torque,  multiplied  by  the  angular 
velocity  w,  gives  the  power  (Eq.  57). 

PROBLEMS 

1.  How  much   work  is  required  to  pump  a  tank  full  of  water  from  a 
40-ft.  well,  the  tank  being  10  ft.  long,  5  ft.  wide,  and  8  ft.  deep,  and  resting 
upon  a  platform  20  ft.  above  the  ground?     The  pipe  enters  at  the  bottom  of 
the  tank.     Assume  that  half  of  the  work  is  done  against  friction,  the  other 
half  against  the  force  of  gravity.     1  cu.  ft.  of  water  weighs  62.4  Ibs.     Sketch 
first. 

2.  A  horse  drawing  a  sled  exerts  a  pull  of  120  Ibs.  upon  the  sled  at  an 
angle  of  20°  with  the  road  bed.     How  much  work  is  required  to  draw  the 
Bled  1/4  mile?     Cos.  20°  =0.94. 

3.  A  10-lb.  force  applied  to  an  18-in.  crank  turns  it  through  4000°. 
How  much  work  is  done? 

4.  A  plow  that  makes  12  furrow  widths  to  the  rod,  i.e.,  which  makes 
16.5-in.  furrows,  requires  an  average  pull  of  300  Ibs.     How  much  work, 
expressed  in  ft  .-Ibs.,  is  done  in  plowing  one  acre? 

6.  What  is  the  potential  energy  of  a  20-kilogram  mass  when  raised  3  ft.? 
Express  the  result  in  ft.-lbs.  and  also  in  ergs. 

6.  What  is  the  kinetic  energy  of  a  200-lb.  projectile  when  its  velocity  is 
1600  ft.  per  sec.? 

7.  If  a  force  of  1961.2  dynes  causes  an  8-gm.  mass  to  slide  slowly  and 
with  uniform  velocity  over  a  level  surface,  what  is  the  coefficient  of  fric- 
tion? 

8.  A  sled  and  rider,  weighing  100  Ibs.,  reaches  the  foot  of  a  hill  64  ft. 
high  with  a  velocity  of  50  ft.  per  sec.     How  much  work  must  have  been 
done  against  friction  on  the  hill? 

9.  At  the  foot  of  the  hill  (Prob.  8)  is  a  level  expanse  of  ice.     Neglecting 
air  friction,  how  far  will  the  sled  (vel.  50  ft.  per  sec.)  travel  on  this  ice 
before  coming  to  rest,  assuming  the  coefficient  of  friction  to  be  0.03? 

10.  How  much  coal  would  be  required   per  acre  in   plowing  the  land 
(Prob.  4)  with  a  steam  plow?     Assume  that  6  Ibs.  of  the  coal  burned  can 
do  1  H.P.-hour  of  work,  and  that  half  of  this  work  is  done  in  pulling  the 
engine,  and  the  other  half  in  pulling  the  plow. 

11.  A  200-lb.  car  A  and  a  50-lb.  car  B  when  at  rest  on  the  same  level 


WORK,  ENERGY  AND  POWER  109 

track  are  connected  by  a  stretched  spring  whose  average  tension  for  3 
seconds  is  2  Ibs.  greater  than  that  necessary  to  overcome  the  friction  of 
running  the  cars.  Find  the  momentum  and  the  kinetic  energy  of  each 
car  at  the  close  of  the  3-sec.  interval. 

12.  What  is  the  average  H.P.  developed  by  the  powder,  if  the  projectile 
(Problem  6)  takes  0.02  sec.  to  reach  the  muzzle,  i.e.,  if  the  pressure  pro- 
duced by  the  powder  acts  upon  the  projectile  for  0.02  sec.? 

13.  What  is  the  average  force  pushing  the  projectile  (Prob.  6)  if  the 
cannon  is  20  ft.  in  length? 

14.  A  runaway  team,  pulling  200  Ibs.,  develops  10  H.P.     How  fast  must 
they  travel? 

15.  How  fast  must  a  400-lb.  bear  climb  a  tree  in  order  to  develop  2  H.P.? 

16.  What  is  the  kinetic  energy  of  a  3-ton  flywheel  when  making  180 
R.P.M.,  if  the  average  diameter  of  its  rim  is  12  ft.?     Assume  the  mass  to 
be  all  in  the  rim. 

17.  What  is  the  cost  of  fuel  for  a  locomotive  for  each  ton  of  freight 
that  it  hauls  1000  miles?     Assume  that  the  average  pull  per  ton  of  the 
loaded  train  is  30  Ibs.,  that  the  train  itself  weighs  as  much  as  its  load,  and 
that  the  locomotive  develops  1  H.P.-hr.  from  each  4  Ibs.  of  coal.     The 
coal  costs  $4.00  per  ton. 

18.  A  horse,  drawing  a  sulky  and  occupant  at  the  rate  of  1  mile  in  2 
min.,  exerts  a  10-lb.  pull  upon  the  sulky.     How  much  more  power  must  the 
horse  furnish  than  if  it  were  to  travel  at  the  same  rate  without  sulky  or 
rider? 

19.  A  steam  engine  being  tested  with  a  Prony  Brake  makes  300  R.P.M. 
and  exerts  at  the  end  of  the  brake  arm,  4  ft.  from  the  axis,  a  force  of  500 
Ibs.     Find  its  H.P. 

20.  Assuming  that  20  per  cent,  of  the  energy  can  be  utilized,  how  many 
H.P.  can  be  obtained  from  a  20-ft.  waterfall  in  a  river  whose  average  width, 
depth,  and  velocity  at  a  certain  point,  are  respectively  50  ft.,  4  ft.,  and  5  ft. 
per  sec.? 

21.  It  is  desired  to  reduce  the  speed  fluctuation  between  successive 
explosions  of  the  10-H.P.  gas  engine  (Sec.  76)  to  1  per  cent,  of  the  average 
speed.     If  the  average  radius  of  the  rim  of  the  flywheel  is  3  ft.,  how  heavy 
must  the  flywheel  be?     Assume  the  mass  to  be  all  in  the  rim.     Also  assume 
in  Eq.  54  that  coi  is  1/2  per  cent,  greater  than  u,  and  that  co2  is  1/2  per 
cent,  less  than  «. 


CHAPTER  VII 
MACHINES 

84.  Machine  Defined. — A  machine  is  usually  a  device  for 
transmitting  power,  though  it  is  sometimes  (e.g.,  the  dynamo)  a 
device  for  transforming  one  kind  of  energy  into  another.  Many 
machines  are  simply  devices  by  means  of  which  a  force,  applied  at 
one  point,  gives  rise  at  some  other  point  to  a  second  force  which, 
in  general,  differs  from  the  first  force  both  in  magnitude  and  direc- 
tion. The  force  applied  to  the  machine  is  called  the  Working 
Force,  and  the  force  against  which  the  machine  works  is  called 
the  Resisting  Force. 

It  is  at  once  apparent  that  whatever  power  is  required  to  over- 
come friction  in  the  machine  itself,  is  power  lost  in  transmission. 
Nevertheless,  transmission  of  power  through  the  machine  may  be 
profitable.  Thus,  in  shelling  corn  with  a  corn  sheller,  the  power 
required  to  separate  the  kernels,  to  mutilate  the  cobs  more  or  less, 
and  to  overcome  friction  of  the  bearings,  must  be  furnished  by  the 
applied  power;  while  if  the  corn  were  shelled  directly  by  hand, 
only  the  power  required  to  separate  the  kernels  would  have  to  be 
applied.  Since  power  is  force  times  velocity  (Eq.  56),  it  is  readily 
seen  that  a  person's  hand  can  apply  a  great  deal  more  power  to  a 
crank  than  it  can  if  pressed  directly  on  the  kernels.  For  both  the 
force  and  the  velocity  may  easily  be  much  greater  in  case  the 
crank  is  used.  Again,  though  a  block  and  tackle  may  transmit 
only  60  per  cent,  of  the  applied  power,  it  is  profitable  to  use  it  in 
lifting  heavy  masses  that  could  not  be  lifted  directly  by  hand.  In 
the  case  of  the  threshing  machine,  the  power  applied  by  the  belt 
from  the  steam  engine  is  transmitted  by  the  threshing  machine  to 
the  cylinder,  to  the  blower,  and  to  numerous  other  parts  of  the 
machine. 

We  shall  here  study  only  what  are  known  as  the  Simple 
Machines.  The  most  complicated  machines  consist  almost 
entirely  in  a  grouping  together  of  the  various  simple  machines 
described  in  the  following  sections.  The  study  of  the  simple 
machines  consists  mainly  in  learning  the  meaning  of  the  efficiency 

no 


MACHINES 


111 


and  the  two  mechanical  advantages  of  each  machine,  and  in  find- 
ing their  numerical  values  from  data  given.  Hence  the  necessity 
for  first  having  a  clear  definition  of  each  of  these  three  terms. 

85.  Mechanical  Advantage  and  Efficiency. — The  Actual  Me- 
chanical Advantage  of  a  machine  is  the  ratio  of  the  resisting  or 
opposing  force  F0,  to  the  force  Fa  applied  to  the  machine,  or 

Act.  Mech.  Adv.  =  F0/Fa 

The  Theoretical  Mechanical  Advantage  is  the  ratio  of  the  dis- 
tance d  through  which  Fa  acts,  to  the  distance  D  through  which 
Fo  acts,  or 

Theor.  Mech.  Adv.  =  d/D 

The  Efficiency  (E)  of  a  machine  is  the  ratio  of  the  useful  work 
W  (i.e.,  F0D)  done  by  the  machine,  to 
the  total  work  Wa  (i.e.,  Fad)  done  upon 
the  machine,  or 

R 


E  = 


Fad 


(57a) 


FIG.  44. 


To  illustrate  the  meaning  of  the 
above  terms,  consider  the  common 
windlass  for  drawing  water  from  a 
well  (Fig.  44).  Let  the  crank,  whose 
length  (K)  is  2  ft.,  rotate  the  drum  of 
6-in.  radius  (r)  upon  which  winds  the 
rope  that  pulls  up  the  bucket  of  water. 
The  hand,  applying  the  force  Fa 
through  the  distance  d,  does  the  work 
Fad  upon  the  machine;  while  the 

bucket,  resisting  with  a  force  F0  (its  weight)  through  a  distance 
D,  has  an  amount  of  work  FoD  done  upon  it  by  the  machine  (the 
windlass). 

From  inspection  we  see  that,  since  R=4r,  d  must  equal  4Z), 
and  the  theoretical  mechanical  advantage  is  therefore  4.  While 
the  theoretical  mechanical  advantage  may  be  found  from  the 
dimensions  as  here  done,  the  actual  mechanical  advantage  must 
always  be  found  from  actual  experiment.  If  the  hand  must  apply 
a  10-lb.  force  to  lift  a  30-lb.  bucket,  the  actual  mechanical  advan- 
tage is  3.  If  the  hand  applying  this  10-lb.  force  moves  2  ft., 
the  bucket  would  rise  6  inches  or  1/2  ft.,  and  the  work  done  upon 


112  MECHANICS  AND  HEAT 

the  machine  would  be  20  f  t.-lbs. ;  while  that  done  by  the  machine 
would  be  15  f  t.-lbs.  (30X1/2).  The  efficiency  (Eq.  57 a)  would 
then  be  15/20,  or  75  per  cent. 

Observe  that  the  efficiency  is  also  equal  to  the  ratio  of  the  two 
mechanical  advantages,  the  actual  to  the  theoretical.  This  is 
always  true.  For,  since  there  is  friction,  the  work  done  by  the 
machine  is  less  than  that  done  upon  it;  i.e.,  the  efficiency 
F0D/Fad,  or  E,  is  less  than  one.  F0D/Fad  =  E  may  be  put  in  the 
form 

F0/Fa  =  EXd/D  '|5£  (58) 


The  left  member  of  this  equation  is  the  actual  mechanical  advan- 
tage, while  the  right  member  is  E  times  the  theoretical  mechanical 
advantage  (note  that  E  is  never  more  than  unity) ;  whence  the 
efficiency  E  is  the  ratio  of  the  two  mechanical  advantages,  which 
was  to  be  proved.  If  it  were  possible  to  entirely  eliminate 
friction,  then  the  work  done  "upon"  and  "by"  the  machine  would 
be  equal  (from  the  conservation  of  energy),  and  therefore  E  would 
be  unity.  Consequently  the  efficiency  would  be  100  per  cent.,  and 
the  theoretical  mechanical  advantage  d/D  would  be  equal  to 
the  actual  mechanical  advantage  F0/Fa.  In  other  words,  the 
theoretical  mechanical  advantage  is  the  ratio  that  we  would  find 
for  F0/Fa  from  the  dimensions  of  the  machine,  neglecting  friction. 
This  condition  of  zero  friction  is  closely  approximated  in  some 
machines. 

86.  The  Simple  Machines. — The  Simple  Machines  are  devices 
used,  as  a  rule,  to  secure  a  large  force  by  the  application  of  a 
smaller  force.     These   machines  are  the  lever,  the  pulley,  the 
wheel  and  axle,  the  inclined  plane,  the  wedge,  and  the  screw. 
Throughout  the  discussion  of  the  simple  machines  the  symbols 
Fa,  F0,  d,  and  D  will  be  employed  in  the  same  sense  as  in  Sec.  85. 
It  may  be  well  to  now  reread  the  last  three  sentences  of  Sec.  84. 
Observe  that  the  theoretical  mechanical  advantage  of  any  simple 
machine,  or  any  combination  of  simple  machines  for  that  matter, 
is  d/D.     Thus,  if  in  the  use  of  any  combination  of  levers  and  pul- 
leys, it  is  observed  that  the  hand  must  move  20  ft.  to  raise  the  load 
1  ft.,  we  know  at  once  that  the  theoretical  mechanical  advantage 
is  20. 

87.  The  Lever. — The  lever  is  a  very  important  and  much  used 
simple  machine.     Indeed,  as  will  be  shown  later,  all  simple  ma- 
chines may  be  divided  into  two  types:  the  lever  type  and  the 


MACHINES  113 

inclined-plane  type.  Though  the  lever  is  usually  a  straight  bar 
free  to  rotate  about  a  support  P,  called  the  fulcrum  or  pivot  point, 
it  may  take  any  form.  Thus  a  bar  bent  at  right  angles  and 
having  the  pivot  at  the  angle  as  shown  at  N(Fig.  45),  is  a  form  of 
lever  that  is  very  widely  used  for  changing  a  vertical  motion  or 
force  to  a  horizontal  one  and  vice  versa. 

There  are  three  general  classes  of  levers,  sometimes  called  1st 
class,  2nd  class,  and  3rd  class,  depending  upon  the  relative  posi- 
tions of  the  fulcrum  or  pivot  P,  and  the  points  A  and  B,  at  which 
are  applied  Fa  and  F0  respectively  (see  Fig.  45).  In  the  class 
shown  at  K,  P  is  between  the  other  two  points;  in  the  class 
shown  at  L,  F0  is  between;  and  in  the  class  shown  at  M,  Fa  is 
between.  In  all  three  cases,  the  applied  torque  about  P  is 
FaXAP,  and,  since  the  lever  is  in  equilibrium  (neglecting  its 


P^  ,~-.~-yP          Fg\}d      ~~B~7D~ P 

A 


FIG.  45. 

weight  and  also  neglecting  friction),  this  torque  must  equal  the 
opposing  torque  due  to  F0,  or  F0XBP.  Hence  FaXAP  = 
F0XBP,  from  which,  noting  that  for  zero  friction  the  two  me- 
chanical advantages  are  equal  (see  close  of  Sec.  85),  we  have 

F       AP 
Theor.  Mech.  Adv.  =  F°  =  gp  (59) 

The  theoretical  mechanical  advantage  may  be  found  in  another 
way.  Let  the  force  Fa  move  point  A  a  distance  d  (all  three  classes) . 
The  point  B  will  then  move  a  distance  D,  and  from  similar  trian- 
gles the  theoretical  mechanical  advantage  d/D  is  seen  to  be  equal 
to  AP/BP,  just  as  in  Eq.  59.  By  measuring  AP  and  BP,  the 
theoretical  mechanical  advantage  is  known.  Thus  if  in  any  case 
AP  equals  3XBP,  it  is  known  at  once  and  without  testing,  that, 


114  MECHANICS  AND  HEAT 

neglecting  friction,  10  Ibs.  applied  at  A  will  lift  30  Ibs.  resting  at 
B.  Friction  in  levers  is  small,  so  that  the  actual  mechanical 
advantage  is  almost  equal  to  the  theoretical,  and  the  efficiency 
is  therefore  nearly  100  per  cent. 

Obviously,  in  using  a  crowbar  to  tear  down  a  building,  the 
resisting  force  F0  is  not  in  general  a  weight  or  load.  Nevertheless, 
since  the  simple  machines  are  very  commonly  used  in  raising 
weights,  it  has  become  customary  to  speak  of  F0  as  the  "load," 
or  the  weight  lifted,  and  Fa  as  the  "force,"  although  both  are 
of  course  forces.  "Resistance"  seems  preferable  to  "load" 
and  we  shall  call  BP  (for  all  three  classes)  the  "resistance  arm," 
and  AP  the  "force  arm."  The  latter  is  sometimes  called  the 
"power  arm,"  but  this  seems  objectionable  inasmuch  as  we  are 
dealing  with  force,  not  power. 

From  the  figure,  it  will  be  seen  that  the  force  arm  may  be  either 
equal  to,  greater  than,  or  less  than  the  resistance  arm  in  levers  of 
the  type  shown  at  K;  while  in  the  type  shown  at  L,  it  is  either 
equal  to,  or  greater  than  the  resistance  arm;  and  in  the  type  shown 
at  M ,  it  is  either  equal  to,  or  less  than,  the  resistance  arm.  Conse- 
quently the  theoretical  mechanical  advantage  (AP/BP)  may 
have  for  the  first-mentioned  type  (K)  any  value;  for  the  next  type 
(L),  one  or  more  than  one;  and  for  the  last  type  (AT),  its  value 
is  one  or  less  than  one.  Observe  that  the  theoretical  mechanical 
advantage  is  always  given  by  the  ratio  of  the  force  arm  to  the 
resistance  arm  (AP/BP},  whatever  the  type  of  lever  may  be. 
The  lever  arm  of  a  force  is  always  measured  from  the  pivot  point. 

The  crowbar,  in  prying  up  a  stone,  may  be  used  as  a  lever 
either  as  shown  at  K  or  at  L.  A  fish  pole  is  used  as  a  lever  of  the 
type  shown  at  M,  if  the  hand  holding  the  large  end  of  the  pole 
remains  at  rest,  while  the  other  hand  moves  up  or  down.  A 
pump  handle  is  usually  a  lever  of  the  type  shown  at  K.  The 
forearm  is  used  as  a  lever  of  type  M  when  bending  the  arm,  and 
type  K  when  straightening  it.  A  pair  of  scissors,  a  pair  of  nut- 
crackers, and  a  pair  of  tweezers  represent,  respectively,  classes 
K,  L,  and  M . 

88.  The  Pulley. — The  theoretical  mechanical  advantage  of 
the  pulley  when  used  as  shown  in  Fig.  46  is  unity.  For  evi- 
dently Fa  must  equal  F0  (neglecting  friction)  in  order  to  make  the 
two  torques  equal.  But  the  theoretical  mechanical  advantage, 
if  we  neglect  friction,  is  F0/Fa  (see  last  three  sentences  of  Sec. 
85).  From  an  actual  test  in  raising  a  load,  it  will  be  found  that 


MACHINES 


115 


Fa  exceeds  F0,  hence  the  actual  mechanical  advantage  is  less 
than  one.  Again,  if  Fa  moves  its  rope  downward  a  distance  d, 
the  weight  W  will  rise  an  equal  distance  D,  and  d/D,  or  the  theo- 
retical mechanical  advantage,  from  this  viewpoint  is  also  seen 
to  be  one. 

Such  a  pulley  does  not  move  up  or  down,  and  is  called  a  fixed 
pulley.  Observe  that  this  pulley  may  be  looked  upon  as  a  lever 
of  the  class  shown  at  K  (Fig.  4.5)  with  equal  arms  r  and  r' .  Al- 
though with  such  a  pulley  F0  is  less  than  the  applied  force  Fa,  the 
greater  ease  of  pulling  downward  instead  of  upward  more  than 
compensates  for  the  loss  of  force. 

The  movable  pulley  is  shown  in  Fig.  47.     With  this  arrangement 


FIG.  46. 


FIG.  47. 


the  pulley  rises  with  the  lifted  weight.  Since  both  ropes  A  and  B 
must  be  equally  tight  (ignoring  friction),  F0  =  2Fa,  or  F0/Fa,  the 
theoretical  mechanical  advantage,  is  2.  This  may  be  seen  in 
another  way  by  considering  point  C  as  the  fulcrum  for  an  instant, 
and  2r  as  the  lever  arm  for  Fa,  and  only  r  as  the  lever  arm  for 
F0.  It  is  also  evident  that  if  rope  B  is  pulled  up  1  ft.  the  weight 
W  will  rise  only  1/2  ft.,  i.e.,  d/D,  the  theoretical  mechanical  advan- 
tage, is  2. 

A  group  of  several  fixed  and  movable  pulleys  arranged  as 
shown  in  Fig.  48  with  a  rope  passing  over  each  pulley  is  called  a 
Block  and  Tackle.  In  practice,  the  pulleys  A  and  B  are  placed 
side  by  side  on  the  same  axle  above;  in  like  manner  C  and  D  are 


116 


MECHANICS  AND  HEAT 


placed  on  one  axle  below.  The  slightly  different  arrangement 
shown  in  the  sketch  is  for  the  purpose  of  showing  more  clearly 
the  separate  parts  of  the  rope.  The  rope  abcde  is  continuous, 
one  end  being  attached  to  the  ring  E  and  the  other  end  being 
held  by  the  hand. 

If  the  applied  force  Fa  on  rope  a  is  say  10  Ibs.,  and  the  pulleys 
are  absolutely  frictionless,  then  the  parts  of  the  rope  b,  c,  d,  and 
e  would  all  be  equally  tight,  and  hence  each  would  exert  an  up- 
ward lift  on  W  of  10  Ibs.,  giving  a  total  of  40  Ibs. 
The  theoretical  mechanical  advantage  is  then 
(neglecting  friction),  F0/Fa  =  40/W  =  4,  or  the 
number  of  supporting  ropes.  Again,  if  W  is 
raised  1  ft.  (D),  each  rope  b,  c,  d,  and  e  will  have 
1  ft.  of  slack,  so  that  a  will  have  to  be  pulled 
down  a  distance  4  ft.  (d)  to  take  up  all  of  the 
slack.  In  other  words,  the  hand  must  move  4 
ft.  to  raise  W  1  ft.  Hence  the  theoretical  mechani- 
cal advantage  from  this  viewpoint  is  4  (i.e. 
d/D=4).  Observe  that  here,  with  a  theoretical 
mechanical  advantage  of  4,  the  weight  moves 
1/4  as  far,  and  hence  1/4  as  fast  as  the  hand. 
This  general  fact  concerning  simple  machines  is 
epitomized  in  the  following  statement:  "What is 
gained  in  force  is  lost  in  speed,  and  vice  versa." 

If  friction  causes  each  pulley  A,  B,  C,  and  D  to 
require  1  Ib.  pull  to  make  it  revolve,  then  if  the 
pull  applied  to  a  were  10  Ibs.,  the  tension  on  6 
would  be  only  9  Ibs. ;  on  c,  8  Ibs. ;  on  d,  7  Ibs. ;  and 
on  e,  6  Ibs.  The  total  lift  exerted  on  W,  i.e.,  F0, 
would  therefore  be  9+8+7+6,  or  30  Ibs.;  hence 
the  actual  mechanical  advantage  F0/Fa  would  be  3. 
Since  the  efficiency  is  the  ratio  of  the  actual  to  the 
theoretical  mechanical  advantage,  it  is  here  3/4,  or 
75  per  cent.  The  efficiency  may  readily  be  found 
in  another  way.  If  the  hand  moves  downward  a  distance  of  4  ft. 
while  exerting  a  force  of  10  Ibs.,  then  the  work  done  upon  the  ma- 
chine is  40  ft.-lbs.,  but  it  has  been  shown  that,  due  to  friction,  this 
force  can  raise  only  30  Ibs.  one  ft.,  i.e.,  the  work  done  by  the  ma- 
chine is  only  30  ft.-lbs.  The  efficiency  is  then  4Q  .  '  ,  —  =75  per 
cent,  as  above.  A  considerably  higher  efficiency  than  this  may 


FIG.  48. 


MACHINES 


117 


be  obtained  if  the  rope  is  very  flexible,  and  if  the  pulley  bearings 
are  smooth  and  well  oiled. 

89.  The  Wheel  and  Axle.—  The  Wheel  and  Axle  (Fig.  49)  con- 
sists of  a  large  wheel  A  of  radius  R  rigidly  attached  to  an  axle  B 
of  radius  r.     A  rope  a  is  attached  to  the  rim  of  the  wheel  and 
wound  around  it  a  few  turns.     Another  rope,  attached  to  the  axle, 
is  secured  to  the  weight  W  that  is  to  be  lifted. 

Viewed  as  a  lever  with  the  axis  as  pivot,  the  theoretical  mechan- 
ical advantage  is  clearly  the  ratio  of  the  two  lever  arms,  or  R/r. 
If  this  ratio  is,  say  5,  the  rope  a  will  have 
to  be  pulled  down  a  distance  (d)  of  5  ft. 
to  lift  the  weight  a  distance  (D)  of  1  ft., 
giving  a  theoretical  mechanical  advantage 
(d/D)  of  5.  If  from  a  test,  the  load  lifted 
is  only  4  times  as  great  as  the  applied  force, 
then  the  actual  mechanical  advantage  is  4, 
and  the  efficiency  (by  Eq.  58)  is  4/5  or  80 
per  cent. 

Observe  that  the  wheel  and  axle  and  the 
windlass  (Fig.  44)  are  exactly  alike  in  prin- 
ciple. It  may  also  be  added  that  practi- 
cally the  only  difference  between  the  cap- 
stan and  the  windlass  is  that  the  drum  is 
vertical  in  the  capstan  and  horizontal  in  the  windlass. 

90.  The  Inclined  Plane.  —  Let  a  rope,  pulling  with  a  force  Fa, 
draw  the  block  E  of  weight  W  up  the  Inclined  Plane  AC  (Fig.  50). 
Resolving  W  into  two  components  (Sec.  19),  the  one  (Fi)  normal, 
the  other  (Fz)  parallel  to  the  plane,  and  noting  that  Fa  equals 
Fz  (if  we  ignore  friction),  we  have  for  the  theoretical  mechanical 
advantage 

=  I/sin  8 


Fia.  49. 


Again,  if  Fa  draws  the  block  from  A  to  C,  it  lifts  the  block  only 
the  vertical  height  BC,  and  the  theoretical  mechanical  advantage, 
d/D,  is  AC/BC,  or  I/sin  6,  as  before.  Observe  that 


AC       slant  height 
--    - 


mi_       TVT 
Theo.  M. 


The  less  steep  the  grade,  the  greater  the  theoretical  mechanical 
advantage,  but  the  block  must  be  drawn  so  much  the  farther  in 
order  to  raise  it  a  given  vertical  distance. 


118 


MECHANICS  AND  HEAT 


If  the  pull  Fa  urging  the  block  up  the  incline,  is  horizontal, 
then,  as  the  block  travels  from  A  to  C  (Fig.  51),  Fa  acts  in  its 
own  direction  through  distance  AB  (i.e.,  d)  and  the  weight  W  is 
raised  the  distance  BC  (i.e.,  D).  Hence  in  this  case 


Theo.  M.  Adv.  = 


AB 


hor.  distance 

r^-r  =  cot0=1/tan0 


BC       vert,  height 


The  equation  just  given  may  be  derived  in  another  way. 
From  Fig.  51  we  see  that  the  pull  on  the  rope,  or  F'a,  must  be  of 
such  magnitude  that  its  component  Fa  parallel  to  AC  shall  equal 
the  force  Fa  of  Fig.  50.  Drawing  F3  equal  to  F'a  but  in  the  oppo- 
site direction,  we  have 


Theo.  M.  Adv.  ---- 


AB 
coie=BC 


The  inclined  plane  is  frequently  used  for  raising  wagon  loads 
and  car  loads  of  material,  for  example,  at  locomotive  coaling 


FIG.  50. 


FIG.  51. 


stations,  and  for  many  other  purposes.  A  train  in  ascending  a 
mountain  utilizes  the  inclined  plane,  by  winding  this  way  and  that 
to  avoid  too  steep  an  incline.  On  a  grade  rising  1  ft.  in  50,  the 
locomotive  must  exert  upon  the  drawbar  a  pull  equal  to  1/50  part 
of  the  weight  of  the  train  in  addition  to  the  force  required  to  over- 
come friction. 

91.  The  Wedge.  —  In  Fig.  52  the  wedge  is  shown  as  used  in 
raising  the  corner  of  a  building.  Fa  represents  the  force  exerted 
upon  the  head  of  the  wedge  by  the  hammer,  and  F0  the  weight  of 
the  corner  of  the  building.  If  Fa  acts  through  the  distance  d 
(the  length  of  the  wedge),  i.e.,  if  Fa  drives  the  wedge  "home," 
then  the  building  will  be  lifted  a  distance  D  (the  thickness  of 
the  wedge),  and  F0  will  resist  through  a  distance  D.  Hence 


Theo.  M.  Adv.  = 


length  of 


D     thickness  of  wedge 


MACHINES 


119 


If  the  hand  exerts  a  force  of  20  Ibs.  upon  a  sledge  hammer 
through  a  distance  of  40  inches,  and  the  hammer  drives  the  wedge 
1  inch,  i.e.,  Fa  acts  through  1  inch,  then  Fa  (average  value)  equals 
20X40  or  800  Ibs.  For,  in  accordance  with  the  conservation  of 
energy,  the  work  done  (force  times  distance)  in  giving  the  ham- 
mer its  motion  must  be  equal  to  the  work  it  does  upon  the  wedge, 
and,  since  the  distance  the  wedge  moves  in  stopping  the  hammer  is 
1/40  as  great  as  the  distance  the  hand  moves  in  starting  it,  the 
force  involved  must  be  40  times  as  great,  or  800  Ibs.  as  already 
found.  If  the  wedge  is  1  in.  thick  and  8  in.  long  it  could,  neglect- 
ing friction,  lift  8X800  or  6400  Ibs.  In  practice,  friction  is  very 
great  in  the  case  of  the  wedge,  so  that  the  weight  lifted  would  be 
very  much  less  than  6400  Ibs.,  say  1600  Ibs.  Accordingly,  if  the 
weight  resting  upon  this  particular  wedge  were  1600  Ibs.,  then 


FIG.  52. 


each  blow  of  the  hammer  would  drive  the  wedge  1  inch  and  raise 
the  building  1/8  in. 

The  actual  mechanical  advantage  of  the  wedge  would  then  be 
1600  Ibs.  -f-  800  Ibs.  or  2,  the  theoretical  mechanical  advantage 
8  in.-i-l  in.  or  8,  and  consequently  the  efficiency  would  be  2-r-8 
or  25  per  cent.  For  wedge  and  hammer  combined,  the  actual 
mechanical  advantage  would  be  1600  Ibs.  -7-20  Ibs.  or  80,  and 
the  theoretical  mechanical  advantage,  40  in. -^  1/8  in.,  or  320. 
Observe  that  the  latter  ratio  (320)  is  the  distance  that  the  hand 
(not  the  wedge)  moves,  divided  by  the  distance  that  the  building 
is  raised.  Thus  we  see  that  the  great  value  of  the  mechanical 
advantage  is  due  to  the  great  force  developed  in  suddenly  stopping 
the  hammer  when  it  strikes  the  wedge,  rather  than  to  the  wedge 


120 


MECHANICS  AND  HEAT 


itself.  A  wedge  would  be  of  little  or  no  value,  if  used  directly, 
that  is,  if  pushed  "home"  by  the  hand. 

If  the  weight  on  the  wedge  were  5  times  as  great  (5  X 1600  Ibs.) 
it  would  require  5  times  as  much  force  to  drive  it,  and  the  hammer 
would  be  stopped  more  suddenly  in  furnishing  this  force.  In 
fact,  the  same  blow  would  drive  the  wedge  1/5  as  far  as  before,  or 
1/5  inch. 

92.  The  Screw. — The  screw  consists  of  a  rod,  usually  of  metal, 
having  upon  its  surface  a  uniform  spiral  groove  and  ridge,  the 
thread.  It  is  a  simple  device  by  which  a  torque  may  develop  a 
very  great  force  in  the  direction  of  the  length  of  the  screw.  For 

example,  by  using  a  wrench  to 
turn  the  nut  on  a  bolt  which 
passes  through  two  beams,  the 
bolt  draws  the  two  beams  for- 
cibly together.  The  principle  of 
the  screw  will  be  readily  under- 
stood from  a  discussion  of  the 
jackscrew,  a  device  much  used 
for  exerting  very  great  forces, 
such  as  in  raising  buildings. 

The  Jackscrew  (Fig.  53)  con- 
sists of  a  screw  S,  free  to  turn  in 
a  threaded  hole  in  the  base  A, 
and  having  at  its  upper  end  a 
hole  through  which  the  rod  BC 

may  be  thrust  as  shown.  Consider  a  force  Fa  applied  at  C  at 
right  angles  to  the  paper  and  directed  inward  (i.e.,  away  from  the 
reader).  Let  it  be  required  to  find  the  weight  F0  that  the  head 
of  the  jackscrew  will  lift.  The  distance  which  the  screws  rise  for 
each  revolution  is  called  the  pitch  p  of  the  screw.  Evidently 
for  each  revolution  of  the  point  C,  the  weight  lifted,  i.e.,  the 
corner  of  the  building,  rises  a  distance  p.  In  doing  this,  however, 
the  force  Fa  applied  to  C  acts  through  a  distance  2nr.  Hence 

o_r 
Theo.  M.  Adv.  =  (d/D)  =  — 

In  the  jackscrew,  friction  is  large,  consequently  the  actual 
mechanical  advantage  is  much  less  than  the  theoretical.  The 
actual  mechanical  advantage  would  be  found  by  dividing  the 


FIG.  53. 


MACHINES 


121 


weight  of  the  corner  of  the  building  (i.e.,  F0)  by  the  force  Fa 
necessary  to  make  C  revolve. 

Both  the  wedge  and  the  jackscrew  involve  the  principle  of  the 
inclined  plane.  This  is  obvious  in  the  case  of  the  wedge.  In  the 
case  of  the  jackscrew,  the  thread  in  the  base  is  really  a  spiral 
inclined  plane  up  which  the  load  virtually  slides.  The  long  rod 
BC  makes  the  mechanical  advantage  much  greater  than  it  is  for 
the  inclined  plane.  Observe  that  all  other  simple  machines 
involve  the  principle  of  the  lever.  Thus  there  are  two  types  of 
simple  machines,  the  inclined-plane  type  and  the  lever  type. 

93.  The  Chain  Hoist  or  Differential  Pulley. — The  Chain 
Hoist  or  Differential  Pulley  (Fig.  54)  is  a  very  convenient  and 


FIG.  54. 


FIG.  55. 


simple  device  for  lifting  heavy  machinery  or  other  heavy  objects. 
It  consists  of  three  pulleys  A,  B,  and  C,  connected  by  an  endless 
chain  of  which  the  portions  c  and  e  bear  the  weight  and  a  and  6 
hang  loose.  The  two  upper  pulleys  A  and  B,  which  differ  slightly 
in  radius,  are  rigidly  fastened  together,  and  each  has  cogs  which 
mesh  with  the  links  of  the  chain.  Designating  the  radius  of  A 
by  r  and  that  of  B  by  r',  let  us  find  the  expression  for  the  theo- 
retical mechanical  advantage. 


122  MECHANICS  AND  HEAT 

Evidently  if  rope  a  is  pulled  down  by  Fa  a  distance  2irr  (i.e.,  d), 
A  will  make  one  revolution,  e  will  be  wound  upon  A  a  distance 
2irr,  and  c  will  be  unwound  from  B  a  distance  27rr'.  Now  the 
latter  distance  is  slightly  smaller  than  the  former,  so  that  the 
total  length  of  e  and  c  is  shortened,  causing  pulley  C,  and  conse- 
quently the  load  W,  to  rise  the  distance  D.  The  above  shortening 
is2irr  —  2-nrr,  or27r(r  —  r'),  and  Crises  only  1/2  this  distance.  Hence 

Theo.  M.  Adv.  =  d/D  =  ^^  =  ^75  (60) 

Eq.  60  shows  that  if  r  and  rf  are  made  nearly  equal,  then  D 
becomes  very  small  and  the  mechanical  advantage,  very  large. 
In  practice,  a  ratio  of  9  to  10  works  very  well,  i.e.,  having,  for 
example,  18  cogs  on  B  and  20  on  A.  In  such  case,  the  above- 
mentioned  shortening  would  be  two  links  per  revolution  (i.e.,  per 
20  links  of  pull),  and  the  rise  D  would  be  one  link,  giving  a  theo- 
retical mechanical  advantage  of  20/1  or  20. 

In  the  chain  hoist  there  is  sufficient  friction  to  hold  the  load 
even  though  the  hand  releases  chain  a.  This  is  a  great  conven- 
ience and  safeguard  in  handling  valuable  machinery.  Likewise 
in  the  case  of  either  the  wedge  or  the  jackscrew,  friction  is  great 
enough  to  enable  the  machine  to  support  the  load  though  the 
applied  force  Fa  is  withdrawn.  This  convenience  compensates 
for  the  low  efficiency  which,  we  have  seen,  is  the  direct  result 
of  a  large  amount  of  friction. 

The  Differential  Wheel  and  Axle  is  very  similar  in  principle  to  the  chain 
hoist.  It  differs  from  the  wheel  and  axle  shown  in  Fig.  49,  in  that  the 
axle  has  a  larger  radius  at  one  end  than  at  the  other. 

If  the  force  Fa  (Fig.  55)  pulls  rope  a  downward  a  distance  (d)  of 
2irR  (R  being  the  radius  of  the  large  wheel),  then,  exactly  as  in  the 
chain  hoist,  rope  e  is  wound  onto  the  large  part  of  the  axle  a  distance 
2irr  and  rope  c  is  unwound  from  the  smaller  part  of  the  axle  a  distance 
27i-r'.  The  shortening  of  ropes  c  and  e  is  2*r— 2*r'  or  2w(r— r'),  and  the 
weight  rises  a  distance  (D)  equal  to  1/2  of  this  distance,  or  ir(r—r'). 

We  thus  have 

rrtl.  -n/r       A    1  d  2irR  2R  /ni\ 

Theo.  M.  Adv.  =  D=ir(r-r^=r^  (61) 

94.  Center  of  Gravity.— The  Center  of  Gravity  (C.G.)  of  a 
body  may  be  defined  as  that  point  at  which  the  entire  weight  of 
the  body  may  be  considered  to  be  concentrated,  so  far  as  the  torque 
developed  by  its  weight  is  concerned.  This  is  equivalent  to  the 


MACHINES 


123 


statement  that  the  C.G.  of  a  body  is  the  point  at  which  the  body 
may  be  supported  in  any  position  without  tending  to  rotate  due 
to  its  weight.  For  its  entire  weight  acts  at  its  C.G.,  and  hence, 
under  these  circumstances,  at  its  point  of  support,  and  therefore 
develops  no  torque.  The  conditions  that  obtain  when  a  body  is 
supported  at  its  C.G.  will  now  be  discussed. 

Let  Fig.  56  represent  a  board  whose  C.G.  is  at  X.  Bore  a  small 
hole  at  X  and  insert  a  rod  as  an  axis.  Through  X  pass  a  vertical 
plane  at  right  angles  to  the  plane  of  the  paper  as  indicated  by  the 
line  AX.  Now  the  positive  torque  due  to  a  mass  particle  mi 
is  its  weight  m\g  times  its  lever  arm  r\.  Proceeding  in  the  same 
way  with  m*  and  all  other  particles 
to  the  left  of  the  line  AX,  and  adding 
all  of  these  minute  torques,  we  ob- 
tain the  total  positive  torque  about  X. 
In  the  same  way  we  find  the  total 
negative  torque  about  the  same  point 
due  to  m3,  w4,  etc.  Since  the  body 
balances  if  supported  at  X,  the  total 
positive  torque  must  equal  the  total 
negative  torque,  and  for  this  reason, 
the  entire  weight  behaves  as  a  single 
downward  pull  W  acting  at  its  C.G. 
This  concept  greatly  simplifies  all  dis- 
cussions and  problems  relating  to  the 

C.G.  of  bodies,  and  will  be  frequently  used.  For  example,  if 
the  rod  is  withdrawn  from  X  and  inserted  at  A,  we  see  at 
once  that  the  downward  pull  W,  and  the  reacting  upward  pull 
of  the  supporting  rod,  will  produce  no  torque,  since  they  lie 
in  the  same  straight  line.  If,  however,  the  rod  is  inserted  at  B, 
the  negative  torque  would  be  TFr,  in  which  r  is  the  horizontal 
distance  between  X  and  B.  If  free  to  do  so,  the  board  would 
rotate  until  B  and  X  were  in  the  same  vertical  line.  In  other 
words,  a  body  always  tends  to  rotate  so  that  its  C.G.  is  directly 
below  the  point  of  support. 

This  tendency  suggests  a  very  simple  means  of  finding  the  C.G. 
of  an  irregular  body,  such  as  C  (Fig.  56) .  Supporting  the  body  at 
some  point  as  D,  determine  the  plumb  line  (shown  dotted). 
Next,  supporting  it  at  E,  determine  another  plumb  line.  The 
intersection  X  of  these  two  lines  is  the  C.G.  of  the  body.  Why? 

Effect  of  C.G.  on  Levers. — If  the  center  of  gravity  of  the  lever 


FIG.  56. 


124  MECHANICS  AND  HEAT 

AB  (sketch  K,  Fig.  45)  of  weight  W,  is  to  the  left  of  P  a  distance  r, 
then  the  weight  of  the  lever  produces  a  positive  torque  Wr, 
which  torque  added  to  that  due  to  Fa,  which  is  also  positive,  must 
equal  that  due  to  F0,  which  is  negative.  Thus  we  see  that  in 
ignoring  the  weight  of  the  lever  we  introduce  into  Eq.  59  a  slight 
error.  This  error  is  negligibly  small  in  the  case  of  lifting  a  heavy 
load  with  a  light  lever.  In  any  given  case  it  can  be  seen  at  a 
glance  whether  this  torque  due  to  the  weight  of  the  lever  helps  or 
opposes  Fa,  remembering  that  all  torques  should  be  computed 
from  the  fulcrum  P. 

95.  Center  of  Mass.  —  The  center  of  mass  (C.M.)  of  any  body  is 
ordinarily  almost  absolutely  coincident  with  its  C.G.  Indeed  the  two 
terms  are  frequently  used  interchangeably.  That  the  two  points  may 
differ  widely  under  some  circumstances,  may  be  seen  by  considering  two 
bodies  of  equal  mass,  one  on  the  surface  of  the  earth,  the  other  1000 
miles  above  the  surface.  Since  the  two  masses  are  equal,  their  common 
center  of  mass  would  be  half  way  between  the  bodies,  or  500  miles  above 
the  earth.  Although  the  two  masses  are  equal,  the  weight  of  the  lower 
body  would  be  roughly  3/2  times  that  of  the  upper  one  (inverse  square 
law),  and  the  center  of  gravity  of  the  two,  which  is  really  the  "center 
of  weight,"  would  be  nearer  the  lower  body.  In  fact,  since  the  weight 
of  the  lower  body  is  3/2  times  that  of  the  upper  one,  its  '-lever  arm," 
measured  from  it  to  the  C.G.,  would  be  2/3  as  great  as  for  the  upper 
body.  The  C.G.  would  therefore  be  400  miles  above  the  earth,  or  100 
miles  lower  than  the  center  of  mass.  As  a  rule,  however,  the  C.M.  of  a 
body  is  practically  coincident  with  its  C.G. 

Center  of  Population.  —  The  center  of  population  of  a  country  is  very 
closely  analogous  to  the  center  of  mass  of  a  body,  and  is  also  a  matter  of 
sufficient  interest  to  warrant  a  brief  discussion.  To  simplify  the  discus- 
sion, let  us  use  an  illustration.  Suppose  that  we  have  found  that  the 
center  of  population  of  the  cities  (only)  of  the  United  States  is  at  Cin- 
cinnati. Through  Cincinnati  draw  a  north  and  south  line  A,  and  an 
east  and  west  line  B.  Now  multiply  the  population  of  each  city  east  of 
line  A  by  its  distance  from  A  and  find  the  sum  of  these  products.  Call 
this  sum  Si.  Next  find  the  similar  sum,  say  $2,  for  all  cities  west  of  A. 
Then  81  =  82.  Proceed  in  exactly  the  same  way  for  all  cities  north  of 
line  B,  obtaining  S3;  and  finally  for  all  cities  south  of  B,  obtaining  84- 


It  may  be  of  interest  to  know  that  the  center  of  population  of  the 
United  States,  counting  all  inhabitants  of  both  city  and  country,  was 
very  close  to  Washington,  D.  C.,  in  1800.  It  has  moved  steadily  west- 
ward, keeping  close  to  the  39th  parallel  of  latitude,  until  in  1900  it  was 
in  Indiana  at  a  point  almost  directly  south  of  Indianapolis  and  west  of 
Cincinnati. 


MACHINES  125 

The  mass  particles  of  a  body  bear  the  same  relation  to  its  center  of 
mass  as  does  the  population  of  the  various  cities  to  the  center  of  popu- 
lation of  them  all.  The  subject  is  further  complicated,  however,  by  the 
fact  that  we  are  dealing  with  three  dimensions  in  the  case  of  a  solid 
body,  so  that  the  distances  must  be  measured  from  three  intersecting 
planes  (compare  the  corner  of  a  box)  instead  of  from  two  intersecting 
lines. 

If  a  rod  of  negligible  weight  connecting  a  4-lb.  ball  M  and  a  1-lb. 
ball  m  (Fig.  57)  receives  a  blow  Fa  at  a  point  1/5  of  its  length  from  the 
larger  ball,  which  point  is  the  center  of  mass,  it  will  be  given  motion  of 
translation,   but   no  rotation.     For,  since   the   two 
"lever  arms"  (distance  from  ball  to  C.M.)  are  inver-       °~  ~     , 
sely  proportional  to  their  respective  masses,  the  balls, 
due   to  their  inertia,  produce  equal  (but  opposing) 
torques  about  their  common  C.M.  when  experiencing 
equal  accelerations.     But  if  the  balls  experience  equal 
acceleration,  the  rod    does  not  rotate.     If   such  a 
body  were  thrown,  the  two  balls  would  revolve  about 
their  common  center    of   mass,   which  point  would         , 
trace  a  smooth  curve.    We  may  extend  this  idea  to     $0=  —i 
any  body  of  any  form.     That  is  to  say,  any  free  body 
is  not  caused  to  rotate  by  a  force  directed  toward  (or         FIG.  57. 
away  from)  its  center  of  mass. 

Let  us  again  look  at  the  problem  in  a  slightly  different  way:  Evi- 
dently the  two  torques  about  the  point  (C.M.)  which  receives  the  blow 
(Fa)  must  be  equal  and  opposite.  These  torques  are  produced  by  the 
inertia  forces  F0  and  F'0  which  M  and  m,  respectively,  develop  in  oppos- 
ing acceleration.  Since  F'0  acts  upon  4  times  as  long  a  lever  arm  (meas- 
ured from  C.M.)  as  does  F0,  it  must  be  1/4  as  large  as  that  force  to 
produce  an  equal  torque,  and  it  will  therefore  impart  to  the  1-lb.  mass 
m  an  acceleration  exactly  equal  to  that  imparted  by  F0  to  the  4-lb.  mass 
M.  If,  however,  the  balls  experience  equal  accelerations  the  rod  will 
not  rotate. 

The  mass  of  the  earth  is  about  80  times  that  of  the  moon,  so  that  the 
moon's  "lever  arm"  (about  their  common  C.M.)  is  80  times  as  long 
as  that  of  the  earth,  and  the  C.M.  of  the  two  bodies  is  therefore  at  a 
point  1/81  of  the  distance  between  them  (about  3000  mi.),  measured 
from  the  earth's  center  toward  the  moon.  Since  the  radius  of  the  earth 
is  about  4000  miles,  we  see  that  the  C.M.  of  the  earth  and  the  moon  is 
about  1000  miles  below  the  surface  of  the  earth  on  the  side  toward  the 
moon.  This  point  travels  once  a  year  around  the  sun  in  a  smooth 
elliptical  path;  while  the  earth  and  the  moon,  revolving  about  it  (the 
C.M.),  have  very  complicated  irregular  paths. 

96.  Stable,  Unstable,  and  Neutral  Equilibrium.— The  Equi- 
librium of  a  body  is  Stahk  if  a  slight  rotation  in  any  direction 


126  MECHANICS  AND  HEAT 

raises  its  center  of  gravity;  Unstable  if  such  rotation  lowers  its 
C.G.;  and  Neutral  if  it  neither  raises  nor  lowers  it.  The  cone, 
placed  on  a  level  plane,  beautifully  illustrates  these  three  kinds 
of  equilibrium. 

When  standing  upon  its  base,  the  cone  represents  stable 
equilibrium,  for  tipping  it  in  any  direction  must  raise  its  C.G. 
To  overturn  it  with  A  as  pivot  point,  its  C.G.  must  rise  a  distance 
h  as  shown  (Fig.  58),  and  the  work  required  (in  foot-pounds) 
A  would  be  the  entire  weight  of  the 

cone  in  pounds  times  h  in  feet,  since 
its  weight  may  be  considered  to  be 
concentrated  at  its  C.G.  If  the  cone 
'is  inverted  and  balanced  upon  its 
apex,  its  equilibrium  is  unstable;  for 
the  least  displacement  in  any  di- 
rection would  lower  its  center  of 
piG  g8  gravity  and  it  would  fall.  Finally, 

the  cone  (also  the  cylinder)  lying 

on  its  side  is  in  neutral  equilibrium,  for  rolling  it  about  on  a 
level  plane  neither  raises  nor  lowers  its  C.G. 

The  equilibrium  of  a  rocking  chair  is  stable  if  the  C.G.  of  the 
chair  and  occupant  is  below  the  center  of  curvature  of  the  rockers. 
For  in  such  case  rocking  either  forward  or  backward  raises  the 
C.G.  Accordingly  a  chair  with  sharply  curved  rockers  is  very  apt 
to  upset,  since  the  center  of  curvature  is  then  low.  To  guard 
against  this,  a  short  portion  of  the  back  end  of  the  rockers  is 
usually  made  straight,  or  better  still,  given  a  slight  reverse 
curvature. 

Equilibrium  on  an  Inclined  Plane. — To  avoid  circumlocution  in 
the  present  discussion  let  us  coin  the  phrase  "Line  of  Centers" 
to  indicate  the  plumb  line  through  the  C.G.  of  a  body.  If  the 
plane  (Fig.  58)  is  inclined,  the  cone  will  be  in  stable  equilibrium 
so  long  as  the  line  of  centers  falls  within  its  base.  The  instant  the 
plane  is  tipped  sufficiently  to  cause  the  line  of  centers  to  fall 
without  its  base,  the  cone  overturns. 

A  loaded  wagon  on  a  hillside  is  in  stable  equilibrium  so  long 
as  the  line  of  centers  (Fig.  59)  falls  within  the  wheel  base. 
Because  of  lurching  caused  by  the  uneven  road  bed,  it  is  unsafe 
to  approach  very  closely  to  this  theoretical  limit.  A  load  of  hay 
is  more  apt  to  upset  on  a  hillside  than  is  a  load  of  coal,  for  two 
reasons.  The  C.G.  is  higher  than  in  the  case  of  the  coal,  and 


MACHINES 


127 


also  the  yielding  of  the  hay  causes  the  C.G.  to  shift  toward  the 
lower  side,  as  from  C  to  D,  so  that  the  line  of  centers  becomes 
DE  (Fig.  59). 

If  the  line  of  centers  falls  well  within  the  base,  a  body  is  not 
easily  upset,  whether  on  an  incline  or  on  a  level  surface.  Manu- 
facturers recognize  this  fact  in  making  broad  bases  for  vases, 
lamps,  portable  machines,  etc.  Ballast  is  placed  deep  in  the 
hold  of  a  ship  in  order  to  lower  its  C.G.  and  thereby  make  it  more 
stable  in  a  rough  sea. 


FIG.  59. 


97.  Weighing  Machines. — The  weighing  of  a  body  is  the 
process  of  comparing  the  pull  of  the  earth  upon  that  body  with 
the  pull  of  the  earth  upon  a  standard  mass,  e.g.,  the  kilogram  or 
the  pound,  or  some  fraction  of  these,  as  the  gram  or  the  ounce, 
etc.  This  comparison  is  not  made  directly  with  the  pull  of  the 
earth  upon  the  standard  kilogram  mass  kept  at  Paris,  or  with  the 
standard  pound  mass  kept  at  London,  but  with  more  or  less  ac- 
curate copies  of  these,  which  may  be  called  secondary  standards. 
We  shall  here  discuss  briefly  the  beam  balance,  steelyard,  spring 
balance,  and  platform  scale.  Each  of  these  weighing  devices, 
except  the  spring  balance,  consists  essentially  of  one  or  more 
levers,  and  in  the  discussion  of  each  a  thorough  understanding 
of  the  lever  will  be  presupposed. 

The  Beam  Balance  consists  essentially  of  a  horizontal  lever  or 
beam,  resting  at  its  middle  point  on  a  "knife-edge"  pivot  of 
agate  or  steel,  and  supporting  a  scalepan  at  each  end,  also  on 


128  MECHANICS  AND  HEAT 

knife-edges.  Usually  a  vertical  pointer  is  rigidly  attached  to  the 
beam.  The  lower  end  of  the  pointer,  moving  over  a  scale,  serves 
to  indicate  whether  the  load  in  one  scalepan  is  slightly  greater 
than  that  in  the  other.  The  body  to  be  weighed  is  placed, 
say,  in  the  left  pan,  and  enough  standard  masses  from  a  set  of 
"weights"  are  placed  in  the  right  pan  to  "balance"  it.  If  too 
much  weight  is  placed  in  the  right  pan,  the  right  end  of  the  beam 
will  dip.  Obviously  if  the  balance  is  sensitive,  a  very  slight  excess 
weight  will  produce  sufficient  dip,  and  consequently  sufficient 
motion  of  the  pointer  to  be  detected.  The  Sensitiveness  of  the 
balance  depends  upon  two  factors,  the  position  of  the  C.G.  of 

the  beam  and  pointer,  and 
the  relative  positions  of  the 
three  knife-edges. 

These  factors  will  now  be 
discussed  in  connection  with 
Fig.  60,  which  is  an  exagger- 
ated diagrammatic  sketch  of 
the  beam  and  pointer  only. 

If  the  C.G.  of  the  beam  and 
GO.  pointer  is  far  below  the  cen- 

tral knife-edge  as  shown,  then 

a  slight  dip  of  the  right  end  of  the  beam  will  cause  the  C.G.  to 
move  to  the  left  a  comparatively  large  distance  r,  and  there- 
fore give  rise  to  a  rather  large  opposing  restoring  torque  equal 
to  the  weight  W  of  the  beam  and  pointer  times  its  lever  arm  r, 
or  a  torque  Wr.  Observe,  as  stated  in  Sec.  94,  that  so  far  as  the 
torque  due  to  the  weight  of  the  beam  and  pointer  is  concerned, 
their  entire  weight  may  be  considered  to  be  at  their  C.G.  From 
the  figure,  we  see  that  if  the  C.G.  were  only  1/2  as  far  below  the 
knife-edge,  then  r  would  be  1/2  as  great,  and  1/2  as  great  ex- 
cess weight  in  the  right  pan  would,  as  far  as  this  factor  is  con- 
cerned, produce  the  same  dip,  and  hence  the  same  deflection 
of  the  pointer  as  before.  Accordingly,  a  sensitive  balance  is  de- 
signed so  that  the  C.G.  is  a  very  short  distance  below  the  cen- 
tral knife-edge,  and  the  smaller  this  distance,  the  more  sensitive 
the  balance. 

Let  us  now  consider  the  second  factor  in  determining  the  sensi- 
tiveness of  a  beam  balance.  If  the  end  knife-edges  are  much 
lower  than  the  middle  one,  as  in  the  figure,  then  the  slight  dip 
shortens  the  lever  arm  r\  upon  which  the  right  pan  acts  by  an 


MACHINES  129 

amount  a  while  at  the  same  time  the  length  of  r2  is  very  slightly 
increased.  Consequently,  under  these  circumstances,  a  compara- 
tively large  restoring  torque  arises,  and  therefore  a  comparatively 
large  excess  weight  in  the  right  pan  will  be  required  to  produce  a 
perceptible  dip  of  the  beam  or  deflection  of  the  pointer.  Hence 
sensitive  balances  have  the  three  knife-edges  in  a  straight  line,  or 
very  nearly  so. 

We  shall  now  slightly  digress  in  observing  that  if  the  three 
knife-edges  represent  in  position  the  three  holes  in  a  two-horse 
"  evener,"  and  if  the  horse  at  each  end  of  the  evener  be  represented 
in  the  figure  as  pulling  downward,  then  the  "ambitious"  horse 
would  have  the  greater  load,  for,  as  just  pointed  out,  the  lower 
end  has  the  shorter  arm.  If  the  horses  are  represented  as 
pulling  upward  in  the  figure,  then  the  horse  that  is  ahead  pulls 
on  the  longer  lever  arm  and  hence  has  the  lighter  load.  This  is 


i  1 1 1 1 1  i  1 1  1 1  i  1 1  1 1  i  1 1 1 1 — >. 

b     /    15  A  25  3JO  ) 


FIG.  01. 

the  usual  condition;  since  the  middle  hole  in  the  evener  is  usually 
slightly  farther  forward  than  the  end  ones. 

The  Steelyard  consists  of  a  metal  bar  A  (Fig.  61),  supported  in 
a  horizontal  position  on  the  knife-edge  B  near  the  heavy  end,  and 
provided  with  a  sliding  weight  S,  and  a  hook  hanging  on  knife- 
edge  C  for  supporting  the  load  W  to  be  weighed.  The  supporting 
hook  //  is  frequently  simply  held  in  the  hand.  In  weighing,  the 
slider  is  moved  farther  out,  thus  increasing  its  lever  arm,  until  it 
"balances"  the  load.  The  weight  of  the  load  is  then  read  from 
the  position  of  the  slider  on  the  scale. 

The  scale  may  be  determined  as  follows:  Remove  W  and  slide 
£  back  and  forth  until  a  "balance"  is  secured.  Mark  this  posi- 
tion of  the  slider  as  the  zero  of  the  scale.  Next  put  in  the  place  of 
W  a  mass  of  known  weight,  say  10  Ibs.,  and  when  a  balance  is 
again  secured  mark  the  new  position  of  the  slider  "10  Ibs."  Lay 
off  the  distance  between  these  two  positions  into  ten  equal  spaces 
and  subdivide  as  desired  the  pound  divisions  thus  formed.  The 


130  MECHANICS  AND  HEAT 

pound  divisions  should  all  be  of  the  same  length.  For,  if  moving 
S  one  division  to  the  right  enables  it  to  balance  1  Ib.  more  at  W, 
then  moving  it  twice  as  far  would  double  the  additional  torque  due 
to  S,  and  hence  enable  it  to  balance  2  Ibs.  more  at  W.  The  same 
scale  may  be  extended  to  the  right  end  of  the  bar. 

The  steelyard  is  made  more  sensitive  by  having  its  C.G.  a 
very  small  distance  below  the  supporting  knife-edge  B,  for  reasons 
already  explained  in  the  discussion  of  the  beam  balance.  This  is 
accomplished  by  having  the  heavy  end  of  the  bar  bent  slightly 
upward,  thereby  raising  its  C.G. 

The  Spring  Balance  consists  essentially  of  a  spiral  steel  spring, 
having  at  its  lower  end  a  hook  for  holding  the  load  to  be  weighed. 
Near  the  lower  end  of  the  spring  a  small  index  moves  past  a  scale, 
and  indicates  by  its  position  the  weight  of  the  load.  Since  the 
spring  obeys  Hooke's  Law  (Sec.  107),  that  is,  since  its  elongation 
is  directly  proportional  to  the  load,  a  scale  of  equal  divisions  is 
used  just  as  with  the  steelyard. 

The  Platform  Scale. — In  the  platform  scale  two  results  must 
be  accomplished;  first,  a  small  "weight"  must  "balance"  the 
load  of  several  tons;  and  second,  the  condition  of  balance  must 
not  depend  upon  what  part  of  the  platform  the  load  is  placed. 
The  first  result  is  accomplished  by  the  use  of  the  Compound 
Lever.  A  Compound  Lever  consists  of  a  combination  of  two  or 
more  levers  so  connected  that  one  lever  is  actuated  by  a  second, 
the  second  by  a  third,  and  so  on.  It  is  easily  seen  that  the 
mechanical  advantage  of  a  compound  lever  is  equal  to  the  product 
of  the  mechanical  advantages  of  its  component  levers  taken 
separately.  Thus,  if  there  are  three  component  levers  whose 
mechanical  advantages  are  respectively  x,  y,  and  z,  then  the 
mechanical  advantage  of  the  compound  lever  formed  by  combin- 
ing them  is  xyz.  The  second  result,  namely,  the  independence 
of  the  position  of  the  load  on  the  platform,  is  attained  by  so  ar- 
ranging the  levers  that  the  mechanical  advantage  is  the  same  for 
all  four  corners,  and  therefore  for  all  points  of  the  platform. 

The  system  of  levers  (only)  of  a  common  type  of  platform  scale 
is  shown  in  Fig.  62  as  viewed  cornerwise  from  an  elevated  posi- 
tion. The  four  levers  EA,  ED,  FB  and  GC  are  beneath  the  plat- 
form (indicated  by  dotted  lines).  These  levers  are  supported 
by  the  foundation  on  the  knife-edges  A,  B,  C,  D,  and  they,  in 
turn,  support  the  platform  on  the  knife-edges  A',  B' ,  C',  and  D'. 
The  point  E  is  connected  by  means  of  the  vertical  rod  EH  with 


MACHINES  131 

the  horizontal  lever  U,  which  lever  is  supported  at  7  as  indicated. 
Finally,  J  is  connected  by  means  of  the  vertical  rod  JK  with  the 
short  arm  KL  of  the  horizontal  lever  (scalebeam)  KM.  A 
"weight,"  which  if  placed  on  the  hanger  N  would  "balance" 
1000  Ibs.  on  the  platform,  is  stamped  1000  Ibs.  To  facilitate 
"balancing,"  the  "slider"  S  (compare  the  steelyard)  may  be 
slid  along  the  suitably  graduated  arm  LM.  If  the  "dead  load" 
on  N  balances  the  platform  when  empty,  then  an  additional 
pound  mass  on  N  will  balance  1000  Ibs.  mass  resting  on  the  plat- 
form, provided  the  mechanical  advantage  of  the  entire  system  of 
levers  is  1000. 

and  if  also  FA=FB=GC=GD,  then 


FIG.  62. 

the  downward  force  at  E,  and  hence  the  reading  of  the  scale- 
beam  above,  will  not  depend  upon  where  the  load  is  placed  on  the 
platform.  This  independence  of  the  position  of  the  load  will  be 
easily  seen  by  assigning  numerical  values  to  the  above  distances. 
Let  the  first  four  distances  each  be  6  in.  and  the  second  four  be 
each  6  ft.,  and  let  EA  and  ED  be  each  18  ft.,  then  if  E  rises  1  in. 
(equals  d),  F  and  G  will  each  rise  1/3- in.,  and  A',  B',  C',  and  D' 
each  1/12  times  1/3,  or  1/36  in.  (i.e.,  D).  Consequently,  the 
mechanical  advantage  d/D  is  1  -r- 1/36  or  36,  and  has  the  same 
value  for  all  four  knife-edges  A',  B',  C"'and  D',  showing  that  the 
recorded  weight  is  independent  of  the  position  of  the  load  on  the 
platform,  which  was  to  be  proved. 


132  MECHANICS  AND  HEAT 

If  JI  =  SHI,  then,  since  the  mechanical  advantage  obtained  by 
lifting  at  E  is  36,  the  mechanical  advantage  at  /  will  be  3  times 
36  or  108.  Finally,  if  LM=10LK,  then  a  downward  pull  at  N 
has  a  mechanical  advantage  of  10  times  108,  or  1080.  In  other 
words,  1  Ib.  at  N  will  balance  1080  Ibs.  placed  anywhere  on  the 
platform. 

In  small  platform  scales,  E  connects  directly  to  the  scalebeam 
above.  In  practice,  the  knife-edges  are  supported  (or  support 
the  load,  as  the  case  may  be)  by  means  of  links,  which  permit 
them  to  yield  in  response  to  sudden  side  wise  jarring,  and  thus 
preserves  their  sharpness  and  hence  the  accuracy  of  the  scale. 

PROBLEMS 

1.  It  is  found  that  with  a  certain  machine  the  applied  force  moves  20  ft. 
to  raise  the  weight  6  in.     What  weight  will  100  Ibs.  applied  force  lift,  assum- 
ing friction  to  be  zero?     If  the  efficiency  is  60  per  cent,  what  will  the  100 
Ibs.  lift?     What  is  the  theoretical  mechanical   advantage  of  the  machine? 
What  is  its  actual  mechanical  advantage? 

2.  If  the  distance  AB  (sketch  L,  Fig.  45)  is  36  in.,  BP  is  6  in.,  and  Fa  is 
100  Ibs.,  what  is  F0?     That  is,  what  weight  can  be  lifted  at  Bl 

3.  A  6-ft.  lever  is  used:  (a)  as  shown  in  sketch  K  (Fig.  45),  and  again  (6)  as 
shown  in  sketch  L,  PB  being  1  ft.  in  each  case.     Find  the  applied  force  nec- 
essary to  lift  1000  Ibs.  at  B  for  each  case.     Explain  why  the  answers  differ. 

4.  What  is  the  theor.  m.  adv.  of  the  block  and  tackle  (Fig.  48)  ?     What 
would  it  be  if  inverted,  in  which  position  pulley  A  would  be  below  and  rope 
a  would  be  pulled  upward? 

5.  Sketch  a  block  and  tackle  giving  a  theor.  m.  adv.  of  3,  of  6,  and  of  7. 

6.  What  applied  force  would  raise  1000  Ibs.  by  using  a  wheel  and  axle, 
if  the  diameter  of  the  wheel  were  4  ft.,  and  that  of  the  axle  6  in.,  (a)  neglect- 
ing friction,  (6)  assuming  90  per  cent,  efficiency? 

7.  A  hammer  drives  a  wedge,  which  is  2  in.  thick  and  1  ft.  in  length,  a 
distance  of  1/2  in.  each  stroke.     The  wedge  supports  a  weight  of  1  ton  and 
the  hand  exerts  upon  the  hammer  an  average  force  of  20  Ibs.  through  a  dis- 
tance of  3  ft.  each  stroke.     What  is  the  theor.  m.  adv.  of  the  wedge?     Of 
both  wedge  and  hammer? 

8.  Find  the  theoretical  and  also  the  actual  mechanical  advantage  of  a 
jackscrew  of  30  per  cent,  efficiency,  whose  screw  has  10  threads  to  the  inch 
and  is  turned  by  a  rod  giving  a  2-ft.  lever  arm. 

9.  Neglecting  friction,  what  pull  will  take  a  200-ton  train  up  a  1  per  cent, 
grade  (i.e.,  1  ft.  rise  in  100  ft.)? 

10.  What  is  the  value  of  the  actual  m.  adv.,  and  also  what  is  the  efficiency 
of  the  combination  mentioned  in  problem  7? 

11.  If  the  jackscrew   (Prob.  8)  is  placed  under  the  lever  at  A  (sketch  L, 
Fig.  45),  what  lift  can  be  exerted  at  B  (of  the  lever)  by  applying  a  50-lb. 
pull  at  the  end  of  the  jackscrew  lever?     Let  lever  arm  BP  be  2  ft.  and  BA,  3  ft. 


MACHINES  133 

12.  What  H.P.  does  the  locomotive  (Prob.  9)  develop  in  pulling  the  train, 
if  its  velocity  is  40  ft.  per  sec.,  and  if  the  work  done  against  friction  equals 
that  done  against  gravity? 

13.  In  a  certain  chain  hoist  the  two  upper  pulleys,  which  are  rigidly 
fastened   together,   have  respectively  22  and  24  cogs,     (a)  What  is  its 
theoretical  mechanical  advantage?     (6)  What  load  could  a  150-lb.  man  lift 
with  it,  assuming  an  efficiency  of  30  per  cent.? 

14.  Let  two  levers,  which  we  shall  designate  as  G  and  H,  be  represented 
respectively  by  the  sketches  K  andL  (Fig.  45),  except  that  G  is  Shifted  to  the 
right  so  that  B  of  G  comes  under  A  of  H,  thus  forming  a  compound  lever. 
Neglecting  friction  (a)  what  downward  force  at  A  of  lever  G  will  lift  1  ton 
at  B  of  lever  H,  provided  AB  and  PB  are  respectively  6  ft.  and  1  ft.  for 
both  levers?     (6)  What  is  the  theoretical  mechanical  advantage  of  G,  and 
of  H,  and  also  of  both  combined?    Sketch  first. 

15.  A  barrel  is  rolled  up  an  incline  20  ft.  in  length  and  6  ft.  in  vertical 
height  by  means  of  a  rope  which  is  fastened  at  the  top  of  the  incline,  then 
passes  over  the  barrel,  and  returns  from  the  upper  &ide  of  the  barrel  in  a 
direction  parallel  to  the  incline.     What  theoretical  mechanical  advantage  is 
obtained  by  a  man  who  pulls  on  the  return  rope? 

16.  A  man,  standing  in  a  bucket,  pulls  himself  out  of  a  well  by  means  of 
a  rope  attached  to  the  bucket  and  then  passing  over  a  pulley  above  and  re- 
turning to  his  hand.     What  theoretical  mechanical  advantage  does  he  have? 

17.  The  drum  of  an  ordinary  capstan  for  house  moving  is  16  in.  in  diame- 
ter, and  the  sweep,  to  which  is  hitched  a  horse  pulling  200  Ibs.,  is    12  ft. 
long.     Find  the  pull  on  the  cable,  assuming  no  friction  in  the  drum  bearings. 

18.  If  in  the  lever  BP,  sketch  M,  Fig.  45,  AB=AP,  what  weight  can  be 
lifted  at  B  if  the  block  and  tackle  shown  in  Fig.  48  lifts  on  A  of  the  lever, 
and  if  the  pull  on  rope  a  of  the  block  and  tackle  is  100  Ibs.?     Neglect  friction. 

19.  The  weight  of  a  24-ft.  timber  is  to  be  borne  equally  by  three  men  who 
are  carrying  it.     One  man  is  at  one  end  of  the  timber  while  the  other  two 
lift  by  means  of  a  crossbar  thrust  under  the  timber.     How  far  from  the  end 
should  the  crossbar  be  placed? 

20.  If  the  lever  AB  (sketch  K,  Fig.  45)  be  a  plank  20  ft.  long  and  weigh- 
ing 100  Ibs.,  and  if  PB  be  2  ft.,  what  downward  force  at  A  will  lift  1000  Ibs. 
at  B,  (a)  if  we  consider  the  weight  of  the  plank?     (&)  If  we  neglect  it? 

21.  A  20-ft.  plank  which  weighs  120  Ibs.  lies  across  a  box  4  ft.  in  width, 
with  one  end  A  projecting  7  ft.  beyond  the  box.     How  near  to  the  end  A  of 
the  plank  can  a  60-lb.  boy  approach  without  upsetting  the  plank?     How 
near  to  the  other  end  may  be  approach? 

22.  How  far  from  the  end  of  the  timber  should  the  crossbar  be  placed 
(Prob.  19)  if  there  are  two  men  lifting  on  each  end  of  it;  one  man  lifting  on 
the  end  of  the  timber  as  before? 

23.  In  Fig.  62,  let  BB'  (etc.)  equal  4  in.,  BF  (etc.)  equal  5  ft.,  AE  (and 
DE)  equal  15  ft.,  HI  =  5  in.,  JI  =20  in.,  LK  =  1 . 5  in.,  and  LM  =  30  in.     What 
weight  at  N  will  balance  2  tons  on  the  platform  of  the  scale? 


PART  II 
PROPERTIES  OF  MATTER 


CHAPTER  VIII 

THE  THREE  STATES  OF  MATTER  AND  THE  GENERAL 
PROPERTIES  OF  MATTER 

98.  The  Three  States  of  Matter. — Matter  exists  in  three  dif- 
ferent states  or  forms:  either  as  a  solid,  as  a  liquid,  or  as  a  gas. 
Liquids  and  gases  have  many  properties  in  common  and  are  some- 
times classed  together  as  fluids. 

We  are  familiar  with  the  general  characteristics  which  distin- 
guish one  form  of  matter  from  another.  Solids  resist  change  of 
size  or  shape;  that  is,  they  resist  compression  or  extension,  and 
distortion  (change  of  shape).  Solids  therefore  have  rigidity, 
a  property  which  is  not  possessed  by  fluids.  Liquids  resist  com- 
pression, but  do  not  appreciably  resist  distortion  or  extension. 
For  these  reasons  a  quantity  of  liquid  assumes  the  form  of  the 
containing  vessel.  Gases  are  easily  compressed,  offer  no  resist- 
ance to  distortion,  and  tend  to  expand  indefinitely.  Thus  a 
trace  of  gas  introduced  into  a  vacuous  space,  for  example,  the 
exhausted  receiver  of  an  air  pump,  will  immediately  expand  and 
fill  the  entire  space.  Most  substances  change  from  the  solid  to 
the  liquid  state  when  sufficiently  heated;  thus  ice  changes  to 
water,  and  iron  and  other  metals  melt  when  heated.  If  still 
further  heated,  most  substances  change  from  the  liquid  state  to 
the  gaseous  state;  thus,  when  sufficiently  heated,  water  changes 
to  steam,  and  molten  metals  vaporize.  Indeed,  practically  all 
substances  may  exist  either  in  the  solid,  the  liquid,  or  the  gaseous 
state,  depending  upon  the  temperature  and  in  some  cases  upon  the 
pressure  to  which  the  substance  is  subjected. 

We  commonly  speak  of  a  substance  as  being  a  solid,  a  liquid,  or 
a  gas,  depending  upon  its  state  at  ordinary  temperatures.  Thus 
metals  (except  mercury),  minerals,  wood,  etc.,  are  solids;  mercury, 
water  and  kerosene  are  liquids;  and  air  and  hydrogen  are 
gases.  Mercury  may  be  readily  either  vaporized  or  frozen,  and 
air  can  be  changed  to  a  liquid,  and  this  liquid  air  has  been  frozen 
to  a  solid.  Some  substances,  e.g.,  those  which  are  paste-like  or 
jelly-like,  are  on  the  borderline  and  may  be  called  semifluids,  or 

137 


138  MECHANICS  AND  HEAT 

semisolids.  It  is  interesting  to  note  that  mercury  and  bromine 
are  the  only  elements  which  are  liquid  at  ordinary  temperature. 

99.  Structure  of  Matter. — All  matter,  whatever  its  form,  is 
supposed  to  be  composed  of  minute  particles  called  molecules. 
Thus  iron  (Fe)  is  composed  of  iron  molecules,  chlorine  (Cl)  of 
chlorine  molecules,  and  iron  chloride  (FeCl2)  of  iron  chloride 
molecules.  These  molecules  are  composed  of  atoms — like 
atoms  in  the  case  of  an  element,  for  example,  iron,  and  unlike 
atoms  in  the  case  of  compounds.  Thus,  the  iron  chloride  mole- 
cule (FeCl2)  consists  of  one  atom  of  iron  (Fe)  and  two  atoms  of 
chlorine  (Cl). 

Molecular  Freedom. — In  the  case  of  a  solid,  the  molecules  that 
compose  it  do  not  easily  move  with  respect  to  each  other.  This 
gives  the  solid  rigidity  which  causes  it  to  resist  any  force  tending 
to  make  it  change  its  shape.  In  liquids,  the  molecules  glide 
readily  over  each  other,  so  that  a  liquid  immediately  assumes  the 
shape  of  the  containing  vessel.  In  gases,  the  molecules  have  even 
greater  freedom  than  in  liquids,  and  they  also  tend  to  separate 
so  as  to  permeate  the  entire  available  space  as  mentioned  in  the 
preceding  section. 

Divisibility  of  Matter. — Any  portion  of  any  substance  may  be 
divided  and  subdivided  almost  without  limit  by  mechanical 
means,(but  so  long  as  the  molecule  remains  intact,  the  substance  is 
unchanged  chemically.  Thus  common  salt  (NaCl),  which  is  a 
compound  of  the  metal  sodium  (Na)  and  chlorine,  may  be  ground 
finer  and  finer  until  it  is  in  the  form  of  a  very  fine  dust,  and  still 
preserve  the  salty  taste.  This  powdered  salt  may  be  used  for 
curing  meats,  and  chemically  it  behaves  in  every  way  like  the 
unpowdered  salt.  If,  however,  through  some  chemical  change  the 
molecule  is  broken  up  into  its  separate  atoms,  namely,  sodium  and 
chlorine,  it  no  longer  exists  as  salt,  nor  has  it  the  characteristics 
of  salt.  Hence  we  may  say  that  the  molecule  is  the  smallest 
portion  of  a  substance  which  can  exist  and  retain  its  original 
chemical  characteristics.  Certain  phenomena  indicate  that  the 
molecule  is  very  small — probably  a  small  fraction  of  one-mil- 
lionth of  an  inch  in  diameter. 

The  Kinetic  Theory  of  Matter. — According  to  this  theory,  which 
is  generally  accepted,  the  molecules  of  any  substance,  whether  in 
the  solid,  the  liquid,  or  the  gaseous  state,  are  in  continual  to-and- 
fro  vibration.  In  solids,  the  molecule  must  remain  in  one  place 
and  vibrate;  in  liquids  and  gases  it  may  wander  about  while 

. 


THE  THREE  STATES  OF  MATTER  139 

maintaining  its  vibration.  This  vibratory  motion  of  translation 
is  supposed  to  give  rise  to  the  diffusion  of  liquids  and  gases  (Sees. 
112  and  131). 

Form  certain  experimental  facts,  a  discussion  of  which  is 
beyond  the  scope  of  this  work,  the  average  distance  through  which 
a  hydrogen  molecule  vibrates,  or  its  "mean  free  path,'!  is  esti- 
mated to  be  about  7/1,000,000  inch,  if  the  hydrogen  is  under  ordi- 
nary atmospheric  pressure,  and  at  the  temperature  of  melting 
ice.  This  distance  is  smaller  for  the  molecules  of  other  gases, 
and  presumably  very  much  smaller  in  the  case  of  liquids  and  solids. 
As  a  body  is  heated,  these  vibrations  become  more  violent.  This 
subject  will  be  further  discussed  under  "The  Nature  of  Heat" 
(Sec.  160),  and  the  "Kinetic  Theory  of  Gases"  (Sec.  171). 

Brownian  Motion. — About  80  years  ago,  Robert  Brown  dis- 
covered that  small  (microscopic)  particles  of  either  organic  or 
inorganic  matter,  held  in  suspension  in  a  liquid,  exhibited  slight 
but  rapid  to-and-fro  movements.  In  accordance  with  the  ki- 
netic theory  of  matter,  these  movements  may  be  attributed  to 
molecular  bombardment  of  the  particles. 

100.  Conservation  of  Matter. — In  spite  of  prolonged  research 
to  prove  the  contrary,  it  still  seems  to  be  an  established  fact  that 
matter  can  be  neither  created  nor  destroyed.     If  several  chemicals 
are  recombined  to  form  a  new  compound,  it  will  be  found  that, 
the  weight,  and  therefore  the  mass,  of  the  compound  so  formed,  is 
the  same  as  before  combination.     When  a  substance  is  burned, 
the  combined  mass  of  the  substance  and  the  oxygen  used  in  com- 
bustion is  exactly  equal  to  the  combined  mass  of  ash  and  the 
gaseous  products  of  combustion.     When  water  freezes,  its  den- 
sity   changes,   but  its   mass   does  not   change.     Matter  then, 
like   energy,   may  be  transformed  but  neither  destroyed  nor 
created. 

101.  General  Properties  of  Matter. — There  are  certain  proper- 
ties, common  to  all  three  forms  of  matter,  which  are  termed 
General  Properties.     Important  among  these  are  mass,  volume, 
density,    gravitational    attraction,    intermolecular   attraction,    and 
elasticity. 

As  a  rule,  any  portion  of  matter  has  a  definite  mass  and  a  defi- 
nite volume.  Dividing  the  mass  of  a  body  by  its  volume  gives  its 
Density,  i.e., 


140 


MECHANICS  AND  HEAT 


In  the  case  of  a  solid  of  regular  form,  its  volume  may  be  deter- 
mined from  measurement  of  its  dimensions.  Its  mass,  whatever 
its  shape,  would  be  obtained  by  weighing.  (For  the  method  of 
obtaining  the  density  of  an  irregular  solid  see  Sec.  122.)  Below 
are  given  the  densities  of  several  substances  in  the  C.G.S. 
system,  i.e.,  in  grams  per  cubic  centimeter.  The  density  of 
water  is  practically  1  gm.  per  cm.3,  or,  in  the  British  system,  62.4 
Ibs.  per  cu.  ft.  Densities  are  usually  expressed  in  the  C.G.S. 
system. 

AVERAGE  DENSITIES  OF  A  FEW  SUBSTANCES 


Solids  (gm 

per  cm.3)      Liquids  (gm. 

per  cm.3) 

Gases  (gm.  per  cm.3) 

Gold  
Lead 

..    19.30 
11  36 

Mercury  
Bromine  
Glycerine.  .  .  . 
Milk  (whole)  . 

Sea-water  
Water,  4°  C.. 
Cream,  about 
Alcohol  .  .  . 

...    13.60 
...     3.15 
.  ..      1.26 
1.028  to 
1.035 
...      1  .  025 
.  ..      1.00 
...      1.00 
.     0.80 

Chlorine  
Carbon  dioxide. 
Oxygen  
Air  
Nitrogen  
Marsh  gas  
Steam,  100°  C.. 
Hydrogen  

0.0032 
0.002 
0.0014 
0.0013 
0.00125 
.0.0007 
0  .  0006 
0.00009 

Silver  
Iron  
Marble.... 
Aluminum. 
Ice  
Cork  

..   10.53 

..      7.80 
..     2.75 
.  .     2  .  60 
..     0.917 
..     0.25 

In  general,  metals  are  very  dense,  as  the  table  shows.  Liquids 
are  less  dense,  and  gases  have  very  small  densities.  Ice  floats  in 
water,  from  which  it  appears  that  the  density  of  water  decreases 
when  it  changes  to  the  solid  state.  Paraffine,  on  the  contrary, 
becomes  more  dense  when  it  solidifies.  The  densities  of  different 
specimens  of  the  same  substance  usually  differ  slightly.  The 
approximate  values  of  those  in  italics  should  be  memorized.  With 
the  exception  of  steam,  the  densities  given  for  the  gases  refer  in 
each  case  to  the  density  of  the  gas  when  at  0°  C.  and  under  stand- 
ard atmospheric  pressure  (Sec.  136). 

Solids,  liquids,  and  gases  all  have  weight,  which  shows  that 
gravitational  attraction  acts  between  them  and  the  earth.  The 
other  two  general  properties,  inter  molecular  attraction  and  elastic- 
ity, will  be  discussed  in  the  following  sections. 

102.  Intermolecular  Attraction  and  the  Phenomena  to  Which 
1  gives  Rise. — It  requires  a  very  great  force  to  pull  a  metal  bar 
in  two,  because  of  the  Intermolecular  Attraction  of  its  molecules. 
If,  however,  the  ends  of  the  bars  are  now  carefully  squared  and 
then  firmly  pressed  together,  it  will  be  found  upon  removing  the 
pressure  that  a  very  slight  force  will  separate  them.  This  ex- 
periment shows  that  this  molecular  force,  which  is  called  Cohesion, 


THE  THREE  STATES  OF  MATTER  141 

and  which  gives  a  metal  or  any  other  substance  its  tensile 
strength,  acts  through  very  small  distances.  Two  freshly  cleaned 
surfaces  of  lead  cohere  rather  strongly  after  being  pressed  firmly 
together.  The  fact  that  lead  is  a  soft  metal,  permits  the  two 
surfaces  to  be  forced  into  more  intimate  contact,  so  that  the 
molecular  forces  come  into  play. 

By  gently  hammering  gold  foil  into  a  tooth  cavity,  the  dentist 
produces  a  solid  gold  filling.  Gold  is  not  only  a  fairly  soft  metal 
but  it  also  does  not  readily  tarnish.  Because  of  these  two  proper- 
ties, the  molecules  of  the  successive  layers  of  foil  are  very  readily 
brought  into  intimate  contact,  and  therefore  unite. 

Welding. — In  welding  together  two  pieces  of  iron,  both  pieces 
are  heated  to  make  them  soft,  and  they  are  then  hammered  to- 
gether to  make  them  unite.  The  "flux"  used  prevents  oxidation 
in  part,  and  also  floats  away  from  between  the  two  surfaces  what- 
ever scale  or  oxide  does  form,  thus  insuring  intimate  contact 
between  them. 

At  is  cohesion  that  enables  the  molecules  of  a  liquid  to  cling 
jfogether  and  form  drops.  This  will  be  further  considered  under 
\/"  Surface  Tension"  (Sec.  124).  If  a  clean  glass  rod  is  dipped  into 
water  and  then  withdrawn,  a  drop  of  water  adheres  to  it.  Obvi- 
ously the  weight  of  the  drop  of  water  is  sustained  by  the  molecular 
attraction  between  the  glass  molecules  and  the  water  molecules. 
This  force  is  called  Adhesion,  whereas  the  force  which  holds  the 
drop  together  is  Cohesion  as  already  stated.  That  is,  the  force 
of  cohesion  is  exerted  between  like  molecules,  adhesion  between 
unlike  molecules. 

Two  pieces  of  wood  may  be  held  firmly  together  by  means  of 
glue.  One  surface  of  the  thin  layer  of  glue  adheres  to  one  piece 
of  wood,  and  the  opposite  surface  adheres  to  the  other.  When 
the  two  pieces  of  wood  are  torn  apart,  the  line  of  fracture  will  .'> 
occur  at  the  weakest  place.  If  the  fracture  occurs  between  glue 
and  wood  in  such  a  way  that  no  glue  adheres  to  the  wood,  then 
the  adhesion  between  glue  and  wood  is  weaker  than  the  cohesion 
of  either  substance.  If  the  layer  of  glue  is  torn  apart  so  that 
a  portion  of  it  adheres  to  each  piece  of  wood,  then  cohesion 
for  glue  is  weaker  than  adhesion  between  glue  and  wood.  Fi- 
nally, if  portions  of  the  wood  are  torn  out  because  of  adhering  to 
the  glue,  which  often  happens,  it  shows  that  the  adhesion  be- 
tween glue  and  wood  is  stronger  than  the  cohesion  of  wood  (at 
that  point). 

' 


142  MECHANICS  AND  HEAT 

As  a  rule,  cohesion  is  stronger  than  adhesion.  The  adhesion 
between  the  layer  of  gelatine  and  the  glass  of  a  photographic 
plate  furnishes  a  striking  exception  to  this  rule.  Sometimes,  in 
becoming  very  dry,  this  gelatine  film  shrinks  with  sufficient  force 
to  tear  itself  loose  from  the  glass  at  some  points,  while  at  other 
points  bits  of  the  glass  are  torn  out,  leaving  the  glass  noticeably 
rough  to  the  touch.  A  thin  layer  of  fish  glue  spread  upon  a 
carefully  cleaned  glass  plate  produces,  as  it  dries,  a  similar  and 
even  more  marked  effect. 

103.  Elasticity,  General  Discussion. — When  a  force  is  applied 
to  a  solid  body  it  always  produces  some  change  either  in  its  length, 
„.;  its  volume,  or  its  shape.  The  tendency  to  resume  the  original 
condition  upon  removal  of  the  applied  force  is  called  Elasticity. 
When  a  metal  bar  is  slightly  stretched  by  a  force,  it  resumes  its 
original  length  upon  removal  of  the  force,  by  virtue  of  its  Tensile 
Elasticity.  If  the  bar  is  twisted,  its  recovery  upon  removal  of 
the  applied  torque  is  due  to  its  Elasticity  of  Torsion,  Rigidity,  or 
Shearing,  as  it  is  variously  termed.  If  the  bar  is  subjected  to 
enormous  hydrostatic  pressure  on  all  sides,  its  volume  decreases 
slightly.  Upon  removal  of  the  pressure,  the  tendency  to  imme- 
diately resume  its  original  volume  is  due  to  the  Volume  Elasticity 
of  the  metal  of  which  the  bar  is  made. 

If,  upon  removal  of  the  distorting  force,  the  body  regains 
immediately  and  completely  its  original  shape  or  size,  it  is  said 
to  be  perfectly  elastic.  Liquids  and  gases  are  perfectly  elastic, 
but  no  solids  are.  Ivory,  glass,  and  steel  are  more  nearly  per- 
fectly elastic  than  any  other  common  solid  substances.  Such 
substances  as  putty  have  practically  no  tendency  to  recover  from 
a  distortion  and  are  therefore  called  inelastic.  They  are  also 
called  plastic,  which  distinguishes  them  from  brittle  inelastic 
substances  such  as  chalk. 

Through  wide  ranges,  most  elastic  substances  are  distorted  in 
proportion  to  the  applied  or  distorting  force,  e.g.,  .doubling  the 
force  produces  twice  as  great  stretch,  twist,  or  shrinkage  in 
volutne,  as  the  case  may  be.  Such  substances  are  said  to  obey 
Hooke's  law  (Sec.  107). 

Any  change  in  the  shape  of  a  body  must  entail  a  change  in  the 
relative  positions  of  its  molecules,  hence  elasticity  of  shape  or 
rigidity  may  be  considered  to  be  due  primarily  to  the  tendency 
of  the  molecules  to  resume  their  former  relative  positions.  The 
resistance  which  the  molecules  offer  to  being  crowded  more  closely 


THE  THREE  STATES  OF  MATTER  143 

together,  or  rather  their  tendency  to  again  spring  apart,  gives 
rise  to  volume  elasticity. 

Elasticity  is  one  of  the  most  important  properties  of  substances, 
and  for  this  reason  it  has  been  very  much  studied.  The  subject 
will  be  taken  up  more  in  detail  in  subsequent  chapters,  especially 
under  "Properties  of  Solids."  For  a  more  complete  study  the 
reader  is  referred  to  advanced  works  on  Physics  or  Mechanics, 
some  of  which  are  mentioned  in  the  preface. 

PROBLEMS 

1.  By  the  use  of  the  table,  find  the  densities  of  air,  sea-water,  mercury, 
and  gold  in  the  British  system 

2.  A  rectangular  block  of  wood  4  in.  X  2  in.  X 1/2  in.  weighs  44  gm.     Find 
its  density. 

3.  Find  the  weight  of  1/2  mi.  of  1/8-in.  iron  wire. 

4.  A  cylindrical  metal  bar  1  cm.  in  diameter  and  20  cm.  in  length  weighs 
165.3  gm.     Of  what  metal  is  it  composed?     What  is  its  density? 

6.  Find  the  mass  of  a  cubic  yard  of  each  of  the  following  substances: 
hydrogen,  air,  water,  ice.  A  cubic  foot  of  water  weighs  62.4  Ibs.  . 

6.  How  many  cubic  feet  of  ice  will  50  gal.  of  water  form  upon  freezing? 
Water  weighs  very  closely  62.4  Ibs.  per  cu.  ft.,  or  8 . 33  Ibs.  per  gal. 

7.  A  hollow  iron  sphere  10  cm.  in  diameter  weighs  3  kilos.     What  is  the 
volume  of  the  cavity  within  it? 

8.  Apiece  of  brass  has  a  density  of  8.4  gm.  per  cm.3     Assuming  that  the 
volume  of  the  brass  is  exactly  equal  to  the  sum  of  the  volumes  of  copper  and 
zinc  that  compose  it,  what  percentage  of  the  brass,  by  volume,  is  zinc? 
Density  of  copper  is  8.92,  zinc  7.2  gm.  per  cm.3     Suggestion:    Represent 
by  x  the  fractional  part  that  is  zinc. 

9.  From  the  answer  to  problem  8,  find  what  percentage  of  the  brass  by 
weight  is  zinc. 


CHAPTER  IX 
PROPERTIES  OF  SOLIDS 

104.  Properties  Enumerated  and  Defined. — The  following 
properties  are  obviously  peculiar  to  solids:  hardness,  brittleness, 
malleability,  ductility,  tenacity  or  tensile  strength,  and  shearing 
elasticity. 

Hardness  and  brittleness  often  go  hand  in  hand.  Thus  steel 
when  tempered  "glass  hard"  is  brittle.  Glass  is  both  hard  and 
brittle.  Chalk,  however,  is  brittle  but  not  hard.  Brittleness 
may  be  defined  as  the  property  of  yielding  very  little  before 
breaking.  Thus  glass  or  chalk  cannot  be  bent,  twisted,  or 
elongated  appreciably  before  breaking,  and  are  therefore  brittle. 

If  a  substance  may  be  made  to  scratch  another,  but  cannot  be 
scratched  by  it,  then  the  former  substance  is  Harder  than  the  latter. 
Ten  substances,  with  diamond  at  the  head  of  the  list,  sapphire 
next  as  9,  and  talc  at  the  bottom  of  the  list  as  1,  have  been  used 
as  a  "  scale  of  hardness."  If  a  certain  substance  may  bescratched 
by  diamond  as  readily  as  it  can  be  made  to  scratch  sapphire, 
then  the  substance  is  9.5  in  the  scale  of  hardness. 

Malleability  is  that  property  of  a  solid  by  virtue  of  which  it  may 
be  hammered  into  thin  sheets.  Gold  is  very  malleable,  indeed 
it  is  the  most  malleable  known  substance.  By  placing  a  thin 
sheet  of  gold  between  two  sheets  of  "gold  beater's  skin"  it 
may  be  hammered  into  foil  about  1/200000  inch  thick.  Lead 
is  malleable.  Iron  becomes  quite  malleable  when  heated  to 
a  white  heat.  Wrought  iron  is  slightly  malleable  at  ordinary 
temperatures. 

Ductility  is  that  property  of  a  metal  which  enables  it  to  be 
drawn  out  into  the  form  of  a  fine  wire.  Brass,  copper,  iron  and 
platinum  are  very  ductile.  Although  lead  is  malleable,  it  is  not 
strong  enough  to  be  very  ductile. 

The  Tenacity  or  tensile  strength  of  a  metal  or  other  substance, 
depends,  as  stated  in  Sec.  102,  upon  the  cohesive  force  between  its 
molecules.  Iron  has  a  large  tensile  strength — from  40,000  to 
60,000  Ibs.  per  sq.  in.  Copper  and  lead  have  relatively  low  ten- 
sile strengths. 

144 


PROPERTIES  OF  SOLIDS  145 

106.  Elasticity,  Elastic  Limit,  and  Elastic  Fatigue  of  Solids. — 
If  several  balls,  made  of  different  metals,  are  successively  dropped 
upon  an  anvil  from  a  height  of  a  few  inches,  it  will  be  found  that 
the  first  rebound  carries  the  steel  ball  nearly  to  the  height  from 
which  it  was  dropped.  The  brass  ball  rebounds  less,  and  the  iron 
one  still  less  than  the  brass  one.  The  lead  ball  does  not  rebound, 
but  merely  flattens  slightly  where  it  strikes  the  anvil.  Ivory 
rebounds  better  than  steel.  The  sudden  stopping  of  the  ball  by 
the  anvil  requires  a  large  force  (F  =  Ma),  which  flattens  the 
ball  in  each  case.  If  the  material  is  elastic,  however,  the  flat- 
tened portion  springs  out  again  into  the  spherical  form  as  soon  as 
the  motion  of  the  ball  is  stopped,  and  in  so  doing  throws  the  ball 
into  the  air.  If  the  ball  and  anvil  were  both  perfectly  elastic  the 
first  rebound  would  bring  the  ball  back  to  the  point  from  which 
it  was  dropped.  This  is  a  very  simple,  rough  test  of  elasticity. 
If  the  ball  were  perfectly  elastic,  the  average  force  required  to 
flatten  it  would  be  exactly  equal  to  the  average  force  with  which 
it  would  tend  to  restore  its  spherical  shape.  Obviously,  these 
two  forces  would  each  act  through  the  same  distance,  hence,  the 
work  of  flattening  and  the  work  of  restoring  would  be  equal. 
But  the  former  work  is  equal  to — is  in  fact  due  to — the  potential 
energy  of  the  ball  in  its  original  position,  and  the  latter  work  is 
used  in  throwing  the  ball  back  to  the  height  of  the  first  rebound. 
Accordingly,  this  height  should  be  equal  to  the  distance  of  fall. 
Because  of  molecular  friction,  the  above  restoring  force  is  smaller 
than  the  flattening  force  even  in  the  case  of  the  ivory  ball,  which 
accounts  for  its  failure  to  rebound  to  the  original  height. 

If  a  straight  spring  is  moderately  bent  for  a  short  time  and  is 
then  slowly  released  (to  prevent  vibration),  it  returns  to  its 
original  straight  condition.  If,  however,  it  is  moderately  bent 
and  left  for  years  in  this  bent  condition  and  is  then  slowly  released, 
it  will  immediately  become  nearly  straight,  and  then  very  slowly 
recover  until  it  becomes  practically  straight.  It  might  be  said 
that  the  steel  becomes  "fatigued"  from  being  bent  for  so  long  a 
time.  Accordingly,  it  is  said  to  be  Elastic  Fatigue  of  the  steel 
(see  also  Sec.  108)  which  in  this  case  prevents  the  immediate 
return  of  the  spring  to  its  straight  condition.  Again,  if  the  spring 
is  very  much  bent  and  then  released  it  will  remain  slightly  bent, 
i.e.,  it  will  have  a  slight  permanent  "  set."  In  such  case,  the  steel 
is  said  to  be  "strained"  beyond  the  Elastic  Limit.  All  solids 
are  more  or  less  elastic.  Even  a  lead  bar  if  very  slightly  bent  will 


146  MECHANICS  AND  HEAT 

recover;  but  the  elastic  limit  for  lead  is  very  quickly  reached,  so 
that  if  the  bar  is  appreciably  bent,  it  remains  bent  upon  removal 
of  the  applied  force. 

106.  Tensile  Stress  and  Tensile  Strain.— In  Sec.  103  a 
brief  discussion  of  elasticity  was  given,  in  which  it  was  shown  that 
solids  possess  three  kinds  of  elasticity.  We  shall  now  discuss 
more  in  detail  the  simplest  of  these,  namely,  Tensile  Elasticity, 
and  consider  the  other  two  in  subsequent  sections.  Before  a 
systematic  study  of  the  elastic  properties  of  a  substance  can  be 
made,  it  is  necessary  to  understand  clearly  the  meaning 
of  each  of  the  terms,  Stress,  Strain,  and  Modulus. 

Whenever  an  elastic  body  is  acted  upon  by  a  force 
tending  to  stretch  it,  there  arises  an  equal  internal 
force  tending  to  shorten  it.  See  Principle  of  d'Alem- 


bert  (Sec  .  43).     Thus,  in  Fig.  63,  let  B  be  a  steel  bar 
of  length  L,  say  10  ft.,  and  of  cross  section  A,  say  2 
sq.  in.     When  an  external  force  F  of  20,000  Ibs.  is  ap- 
plied, the  bar  stretches  a  distance  e  (elongation),  say 
0.04  in.     It  is  at  once  evident  that  in  this  stretched 
te  condition,  which  is  also  an  equilibrium  condition,  the 
internal  forces  due  to  which  the  bar  tends  to  resume 
its  normal  length   must  just  equal  the  20,000  Ibs. 
FIG.  63.    which  tends  to  make  the  bar  lengthen;  otherwise  the 
weight  W  would  move  downward  causing  the  stretch 
of  the  bar  to  be  still  further  increased.     This  internal  force  di- 
vided by  the  cross  section  of  the  bar,  in  other  words,  the  force 
per  unit  cross  section,  is  called  the  Tensile  Stress.     But,  since 
the  internal  force  that  arises  is  always  equal  to  the   applied 
force,  we  have 

(62) 


. 

cross  section     A 

20000 
which  here  is  —  „      or  10,000  Ibs.  per  sq.  in. 

The  increase  in  length,  or  the  elongation  e  of  the  bar,  divided 
by  its  original  length,  in  other  words,  the  stretch  per  unit  length, 
is  called  the  Tensile  Strain.  Accordingly,  we  here  have 

Tensne 

A  column,  in  supporting  a  load,  is  subjected  to  a  stress  and 
suffers  a  strain,  both  of  which  are  denned  essentially  as  above. 


PROPERTIES  OF  SOLIDS  147 

The  stress  is  the  load  divided  by  the  cross  section  of  the  column, 
and  the  strain  is  the  decrease  in  length  divided  by  the  original 
length  of  the  column.  It  is  an  observed  fact  that  a  column,  in 
supporting  a  load  in  the  usual  way,  is  decreased  in  length  by  an 
amount  exactly  equal  to  the  stretch  that  it  would  experience  if 
its  upper  end  were  fastened  to  a  support  and  the  same  load  were 
suspended  from  its  lower  end.  In  other  words,  within  certain 
limits,  the  elasticities  of  extension  and  compression  are  alike. 
It  appears,  then,  that  within  certain  limits,  the  molecules  of  an 
elastic  solid  resist  having  their  normal  spacing  decreased  with  the 
same  force  that  they  resist  having  it  increased  a  like  amount. 

107.  Hooke's  Law  and  Young's  Modulus.—  If  the  bar  B 
(Fig.  63)  supports  twice  as  large  a  load  it  will  stretch  twice  as 
much,  and  so  on  for  still  larger  loads,  so  long  as  it  is  not  strained 
beyond  the  elastic  limit.  A  glance  at  the  above  equations  shows 
that  both  the  stress  and  the  strain  must,  then,  increase  directly 
as  does  the  load.  This  being  true,  it  follows  that 
Stress 


which  is  known  as  Hooke's  law.  If  a  substance  is  strained  beyond 
the  elastic  limit  it  does  not  obey  Hooke's  law;  conversely,  if  an 
elastic  body  does  not  obey  Hooke's  law,  it  must  be  strained 
beyond  the  elastic  limit. 

A  spiral  spring  of  steel  obeys  Hooke's  law,  i.e.,  the  elongation 
is  proportional  to  the  load  it  supports.  This  property  is  utilized 
in  the  ordinary  spring  balance  used  in  weighing.  If  a  certain 
torque  twists  a  rod  or  shaft  through  an  angle  of  20°,  and  if 
doubling  the  torque  twists  it  40°,  then  the  rod  or  shaft  follows 
Hooke's  Law  for  that  torque.  If  5  times  as  great  a  torque  twists 
the  rod  say  130°  (instead  of  100°),  it  shows  that  it  is  strained  be- 
yond the  elastic  limit,  since  for  this  larger  torque  it  does  not 
follow  Hooke's  Law. 

The  constant  of  Hooke's  Law  is  called  the  Stretch  Modulus 
or  Young's  Modulus  for  the  substance,  when  applied  to  tensile 
stress  and  tensile  strain,  or 

,     tensile  stress     F/A     FL  . 

Young  s  Modulus  ^  =  tensile^traln  =  "e/L  =  Ae 

Substituting  the  values  used  in  Eqs.  62  and  63,  we  have 

lbs-  <*  •  itt- 


148  MECHANICS  AND  HEAT 

The  above  assumed  stretch  is  about  what  would  be  found  by 
experiment  if  the  bar  were  very  good  steel.  Hence  Young's 
modulus  for  good  steel  is  30,000,000  Ibs.  per  sq.  in.  In  the  metric 
system,  the  force  would  usually  be  expressed  in  dynes  (sometimes 
in  kilograms),  the  distance  in  centimeters,  and  the  cross  section 
in  square  centimeters.  Young's  modulus  for  steel  as  expressed 
in  this  system  is  1.9X1012  dynes  per  cm.2  For  most  substances 
Young's  modulus  is  very  much  smaller  than  for  steel;  in  other 
words,  most  substances  offer  less  resistance  to  stretching  than 
steel  does. 

If,  in  Eq.  64,  A  were  unity,  and  if  e  were  equal  to  L,  i.e.,  if  B  had 
unit  cross  section  and  were  stretched  to  double  its  original  length  (assum- 
ing that  to  be  possible),  then  the  equation  would  reduce  to  E  =  F. 
Hence  Young's  modulus  E  is  numerically  equal  to  the  force  that  would  be 
required  to  stretch  a  bar  of  unit  cross  section  to  twice  its  original  length, 
provided  it  continued  to  follow  Hooke's  law.  Although  a  bar  of  steel, 
or  almost  any  other  substance  except  rubber,  would  break  long  before 
reaching  twice  its  original  length,  still  this  concept  is  useful.  For,  by 
its  use  in  connection  with  the  above  data,  we  see  at  once,  since  a  force 
of  30,000,000  Ibs.  would  double  the  length  of  a  bar  of  1  sq.  in.  cross 
section  (assuming  Hooke's  law  to  hold),  that  a  force  of  30,000  Ibs.,  for 
which  force  Hooke's  law  would  hold,  would  increase  its  length  1/1000  as 
much,  or  1  part  in  1000. 

108.  Yield  Point,  Tensile  Strength,  Breaking  Stress.— If  the 
bar  B  (Fig.  63)  is  made  of  steel,  it  will  be  found  that  as  the  load 
is  increased  the  bar  will  stretch  more  and  more,  in  accordance 
with  Hooke's  law,  until  the  stress  is  about  60,000  Ibs.  per  sq.  in. 
Upon  further  increasing  the  load,  it  will  be  found  that  the  bar 
begins  to  stretch  very  much  more — perhaps  50  times  more — 
than  for  previous  increases  of  like  magnitude.  This  change  in 
the  behavior  of  the  steel,  this  very  great  increase  in  the  strain 
produced  by  a  slight  increase  in  the  stress,  is  due  to  a  yielding 
of  the  molecular  forces,  which  yielding  permits  the  molecules 
to  slide  slightly  with  reference  to  each  other.  We  may  say  for 
this  specimen  of  steel,  that  a  stress  of  60,000  Ibs.  per  sq.  in. 
strains  it  to  the  elastic  limit,  and  that  a  slightly  greater  stress 
brings  it  to  the  Yield  Point. 

As  soon  as  the  yield  point  is  reached,  further  increase  of  load 
causes  the  bar  to  stretch  until  the  elongation  is  25  or  30  per  cent, 
of  the  original  length,  in  the  case  of  soft  steel.  The  maximum 
elongation  for  hard  steel  may  be  as  small  as  1  per  cent.  If  the 


PROPERTIES  OF  SOLIDS  149 

load  is  removed  after  the  yield  point  has  been  passed,  the  bar 
remains  permanently  elongated,  i.e.,  it  has  a  Permanent  Set. 
This  elongation  is  accompanied  by  a  decrease  in  cross  section. 
The  maximum  load  required  to  cause  breaking,  divided  by  the 
original  cross  section,  gives  the  Breaking  Stress  or  Tensile  Strength 
of  the  Steel 

A  slight  difference  in  the  amount  of  carbon  in  steel,  changes  its  elastic 
behavior  very  much.  Thus,  a  certain  specimen  of  steel  containing  0.17 
per  cent,  carbon  had  an  elastic  limit  of  51,000  Ibs.  per  sq.  in.  and  a 
breaking  stress  of  68,000  Ibs.  per  sq.  in.  For  another  specimen,  contain- 
ing 0.82  per  cent,  carbon,  the  elastic  limit  was  68,000  Ibs.  per  sq.  in.,  and 
the  breaking  stress  was  142,000  Ibs.  per  sq.  in.  The  annealing  or  tem- 
pering of  steel  is  also  an  important  factor  in  determining  its  elastic 
properties. 

In  addition  to  iron  and  carbon,  steel  may  contain  various  other  sub- 
stances, important  among  which  are  nickel,  silicon,  and  manganese, 
which  greatly  influence  its  elastic  properties  and  its  hardness,  e\en 
though  present  in  very  small  quantities  (1  to  5  per  cent,  more  or  less). 
A  piano  wire,  having  the  enormous  tensile  strength  of  340,000  Ibs.  per 
sq.  in.,  or  170  tons  per  sq.  in.,  was  found  upon  analysis  to  contain  0.01  per 
cent,  sulphur,  0.018  per  cent,  phosphorous,  0.09  per  cent,  silicon,  0.4  per 
cent,  manganese,  and  0.57  per  cent,  carbon.  Because  of  the  great  com- 
mercial importance  of  steel,  this  brief  statement  concerning  its  composi- 
tion and  elastic  properties  is  made  here.  For  further  discussion  consult 
some  special  engineering  work  on  the  subject,  or  an  encyclopedia,  such 
as  "Americana"  or  "Britannica." 

Factor  of  Safety. — If  steel  is  subjected  to  a  great  many  repeti- 
tions of  stresses  which  are  well  below  its  tensile  strength,  or  even 
below  its  elastic  limit,  it  is  greatly  weakened  thereby,  and  it  may 
finally  break  with  a  load  which  it  would  have  easily  carried 
at  first.  This  weakening  of  material  by  a  great  number  (several 
millions)  of  repetitions  of  a  stress  is  said  to  be  due  to  Elastic 
Fatigue.  (See  also  Sec.  105.)  Of  course  in  any  structure  the 
stress  should  always  be  well  below  the  elastic  limit  for  the  material 
used.  Thus  steel  whose  elastic  limit  is  50,000  Ibs.  per  sq.  in. 
would  rarely  be  subjected  to  stresses  greater  than  25,000  Ibs. 
per  sq.  in.  In  such  case  the  Factor  of  Safety  is  2.  Structures  or 
machine  parts  which  are  exposed  to  vibrations  and  sudden 
stresses  or  shocks,  especially  if  constructed  of  very  hard  steel 
or  other  relatively  brittle  material,  require  a  much  higher  factor 
of  safety.  The  factor  of  safety  also  guards  against  breakage 
(rom  flaws  in  the  material. 


150  MECHANICS  AND  HEAT 

109.  Strength  of  Horizontal  Beams. — If  a  straight  beam  of  wood  or 
metal  (Fig.  64)  of  length  L,  having  a  rectangular  cross  section  of  depth 
h  and  width  a,  is  supported  at  each  end  and  loaded  in  the  middle  as 
shown,  it  will  bend  slightly.  Obviously,  in  the  process  of  bending,  the 
material  near  the  upper  portion  of  the  beam  is  compressed,  while  that 
below  is  stretched.  The  horizontal  layer  of  particles  through  the 
middle  of  the  beam,  that  is,  through  the  line,  BCD,  is  called  the  Neutral 
Plane,  because  this  portion  is  neither  compressed  nor  stretched.  The 
material  at  G  is  stretched  only  1/2  as  much  as  that  at  H,  because  it  is 
only  1/2  as  far  from  the  neutral  plane.  Hence  if  the  load  is  made  too 


FIG.  64. 

great  the  material  at  H,  called  the  "outer  fiber,"  is  the  first  to  be  strained 
to  the  yield  point,  and  when  fracture  occurs,  it  starts  at  this  point. 
It  can  be  shown  by  means  of  advanced  mathematics  that 

<«> 


in  which  d  is  the  deflection  of  the  middle  of  the  beam  produced  by  the 
load  W,  and  E  is  Young's  modulus  for  the  material  of  the  beam.  Eq. 
65  shows  that  the  beam  will  deflect  less,  and  hence  be  stronger  if  placed 
on  edge  than  if  flatwise. 

As  an  illustration,  consider  a  2-in.  by  6-in.  joist  such  as  is  sometimes 
used  to  support  floors.  In  changing  the  joist  from  the  flat  to  the  edge- 
wise position,  we  treble  h  and  make  a  1/3  as  large.  Trebling  h  makes 
h3  27  times  as  large,  consequently  ah3  is  1/3  times  27,  or  9  times  as  large 
as  before.  This  makes  d  1/9  as  large.  In  other  words,  the  beam  would 
require  9  times  as  large  a  load  to  give  the  same  amount  of  bend,  which 
means  that  the  Stiffness  of  the  beam  is  made  9  times  as  great  by  turning 
it  on  edge. 

In  the  edgewise  position,  however,  the  distance  of  the  "outer  fiber" 
(Fig.  64)  from  the  neutral  plane  is  three  times  as  large  as  before,  and 
consequently  a  given  bend  or  deflection  produces  3  times  as  great  a 
strain  on  this  fiber  as  before,  so  that  the  Strength  of  the  beam  is  not  9 
times  as  great,  but  only  3  times  as  great  on  edge  as  flatwise. 


PROPERTIES  OF  SOLIDS  151 

Next  consider  the  effect  on  d  of  variation  in  length,  all  other  quantities 
remaining  the  same.  If  the  beam  is  made  3  times  as  long,  L3  and  hence 
also  d  become  27  times  as  great  as  before.  If  the  beam  is  three  times  as 
long,  it  must  bend  9  times  as  much  (i.e.,  d  must  be  9  times  as  great)  to 
produce  the  same  strain  in  the  material.  For  to  produce  the  same  strain 
in  the  longer  beam,  it  must  bend  to  an  arc  of  the  same  radius  of  curvature 
as  the  shorter  beam.  But,  for  small  arcs,  the  distance  d  from  the  middle 
point  of  the  chord  to  the  middle  point  of  its  arc  varies  approximately 
as  the  square  of  the  length  of  the  chord.  Consequently,  the  strain  is  3 
(not  27)  times  as  great  as  before,  and  the  beam  will  therefore  support 
only  1/3  as  great  a  load  as  before.  This  relation  will  be  clearly  seen 
from  an  application.  Suppose  that  a  pine  beam  4  ft.  long  and  2  in.  by 
4  in.  in  cross  section,  will  support  1000  Ibs.  at  its  center.  Then  if  twice 
as  long  it  will  support  1/2  as  much,  or  500  Ibs.  If  3  times  as  long  it  will 
support  1/3  as  much,  and  so  on. 

To  summarize,  we  may  state  that  for  rectangular  beams  supported  at 
the  end  and  loaded  in  the  middle  (or  supported  in  the  middle  and  loaded 
at  the  ends,  which  amounts  to  the  same  thing),  the  strength  varies  directly 
as  the  first  power  of  the  width  and  as  the  second  power  of  the  depth;  while  it 
varies  inversely  as  the  first  power  of  the  length.  For  such  beams,  the  stiff- 
ness varies  directly  as  the  first  power  of  the  width,  and  as  the  cube  of  the  depth 
(other  things  not  being  varied);  while  it  varies  inversely  as  the  cube  of 
the  length. 

110.  Three  Kinds  of  Elasticity,  of  Stress,  and  of  Strain;  and 
the  Three  Moduli. — In  Sec.  103  it  was  stated  that  a  solid,  for  ex- 
ample a  metal  bar,  may  be  acted  upon  by  forces  in  three  distinct 
ways  bringing  into  play  its  three  elasticities.  Thus  the  metal  bar 
B  (Fig.  65)  of  length  L  and  cross  section  A,  is  acted  upon  by  a  force 
F  which  produces  an  elongation  e.  Upon  removal  of  this  force 
it  returns  to  its  original  length  due  to  tensile  elasticity.  Bi 
illustrates  the  same  bar  acted  upon  by  forces  from  all  sides,  i.e., 
over  its  entire  surface  of  area  A\.  Let  us  suppose  these  forces  to 
be  due  to  hydrostatic  pressure,  which  pressure  causes  a  decrease 
V  in  the  original  volume  (F)  of  the  bar.  As  soon  as  the  pressure 
is  removed,  the  bar  returns  to  its  original  volume  by  virtue  of  its 
volume  elasticity.  B2  illustrates  the  same  bar  again,  this  time  with 
its  lower  surface  fixed.  Consequently  the  force  F  applied  to  its 
upper  surface  of  area  A2  makes  it  slide  or  shear  a  distance  s  with 
respect  to  the  lower  surface.  The  distance  between  the  two  sur- 
faces we  shall  call  d.  Upon  removal  of  the  force  F  the  shear 
disappears  due  to  shearing  elasticity.  In  all  three  cases,  recovery 
upon  removal  of  the  force  is  practically  immediate  and  complete, 
provided  the  bar  has  not  been  strained  beyond  the  elastic  limit. 


152 


MECHANICS  AND  HEAT 


The  Three  Moduli.  —  The  stress  to  which  a  certain  material  is 
subjected,  divided  by  the  resulting  strain,  is  constant  (Hooke's 
Law),  and  this  constant  is  called  the  Modulus  of  Elasticity. 
Since  there  are  three  kinds  of  stress  and  three  kinds  of  strain,  it 
follows  that  there  must  be  three  moduli. 

Stress  is  always  the  total  applied  force  F  divided  by  the  area 
to  which  it  is  applied.  Thus  in  the  first  case  (B),  tensile  stress  is 
F/A,  in  the  second  case  (Bi),  the  hydrostatic  stress  or  volume 

stress  is  F/Aij  while  in  the  third 
z),  the  shearing  stress  is 
In  the  first  case,  the  ten- 
sile strain  is  the  change  in 
length  divided  by  the  original 
length  or  e/L;  in  the  second 
case,  the  volume  strain  is  the 
change  in  volume  divided  by  the 
original  volume,  or  V'/V;  while 
in  the  third  case,  the  shearing 
strain  is  the  distance  sheared 
divided  by  the  distance  be- 
tween the  two  shearing  surfaces, 
or  s/d. 


FIG.  65. 


tensile  stress    F/A 


(64  bis) 


Summarizing,  then,  we  have: 

,     .  _  ,,,         ,  ,     x 

The  mod.  of  Tension  (Young  s  modulus)  = 


mi     TT  i  . .     hydrostatic  pressure 

The  Volume  modulus  (bulk  mod.)  =  -—j-——-—  =  ^ 

(66) 

mi    «.       •  /  .,   \     shearing  stress 

The  Shearing  modulus  (mod.  of  rigidity)  =snearing  strain  = 

(67) 

Observe  that  if  s  is  very  small  with  respect  to  d,  then  s/d  =  6. 
The  angle  d  is  called  the  angle  of  shear.  For  this  reason  the 
shearing  strain  is  usually  called  the  angle  of  shear.  To  illustrate 
shearing,  the  bar  B2  may  be  considered  to  be  made  up  of  a  great 
number  of  horizontal  layers  of  molecules,  a  few  of  which  layers 
are  indicated  in  the  sketch.  Evidently,  when  the  force F  is  applied, 
and  the  bar  is  changed  from  the  rectangular  form  to  the  sheared 
position,  each  layer  is  shifted  to  the  right  a  slight  distance, — s  for 


PROPERTIES  OF  SOLIDS  153 

the  top  layer,  \  s  for  the  middle  layer,  and  so  on.  Further- 
more, each  layer  is  shifted  or  displaced  very  slightly  with  respect 
to  the  next  layer  below  it,  thereby  causing  a  slight  change  in 
the  relative  positions  of  the  molecules  of  successive  layers.  If 
F  is  decreased,  the  tendency  of  the  molecules  to  resume  their 
original  relative  positions  reduces  the  relative  shift  between 
successive  layers,  and  hence  reduces  the  angle  of  shear.  If  F 
is  removed  the  angle  of  shear  becomes  zero,  i.e.,  the  molecules 
completely  return  to  their  normal  relative  positions,  and  the  bar 
again  becomes  rectangular,  provided  it  has  not  been  strained, 
beyond  the  elastic  limit. 

111.  The  Rigidity  of  a  Shaft  and  the  Power  Transmitted.— If 
one  end  A  of  a  shaft  is  clamped  and  the  other  end  B  is  turned 
through  one  revolution  by  some  applied  torque,  the  shaft  is 
said  to  be  twisted  through  an  angle  of  360°.  Evidently  the  layer 
of  molecules  on  the  end  B  has  been  displaced  or  sheared  through 
1  revolution  with  respect  to  the  layer  at  end  A,  through  1/2 
revolution  with  respect  to  the  transverse  layer  through  the  middle 
of  the  shaft,  through  1/4  revolution  with  respect  to  the  layer  1/4 
way  from  B  to  A,  and  so  on.  Indeed  every  transverse  (circular) 
layer  in  the  shaft  is  sheared  slightly  with  respect  to  its  neighbor. 
Obviously  this  shear  is  greatest  for  the  particles  farthest  from  the 
axis  of  the  shaft.  Accordingly  it  is  the  "outer  fibers"  (on  the 
surface  of  the  shaft)  which  first  give  way  when  it  is  twisted  in 
two.  Observe  that  when  a  bolt  is  twisted  in  two,  the  central 
fibers  are  the  last  to  break.  Observe  also  that  fracture  in  this 
case  consists  in  a  shearing  apart  of  adjacent  layers. 

By  knowing  the  values  of  the  shearing  modulus  and  the 
shearing  strength  for  the  steel  used,  and  with  the  aid  of  certain 
formulas,  the  derivation  of  which  requires  a  knowledge  of  ad- 
vanced mathematics,  the  engineer  can  readily  compute  the 
proper  size  of  shaft  for  a  specified  purpose.  The  shaft  must  be 
of  such  size  that  the  maximum  torque  to  which  it  is  to  be  sub- 
jected shall  not  strain  the  outer  fibers  beyond  the  "safe"  limit. 
Although  the  mathematical  treatment  of  this  topic  is  too  com- 
plicated for  an  elementary  work,  it  may  be  stated  that  the 
strength  of  a  shaft,  that  is  the  maximum  torque  which  it  can 
safely  transmit,  varies  as  the  cube  of  its  radius,  while  the  "stiff- 
ness" varies  as  the  4th  power  of  the  radius.  Thus  a  2-inch  shaft 
can  transmit  8  times  as  great  a  torque  as  a  1-in.  shaft;  while,  if 
the  length  of  the  shaft  and  the  applied  torque  are  the  same  for 


154  MECHANICS  AND  HEAT 

both,  the  smaller  shaft  will  be  twisted  through  16  times  as  great 
an  angle  as  the  larger. 

Since  power  is  torque  multiplied  by  the  angular  velocity 
(P=Ta>,  Sec.  83),  it  follows  that  a  given  amount  of  power 
can  be  transmitted  by  1/4  as  great  a  torque,  and  hence  by 
1/4  as  strong  a  shaft  by  making  the  angular  velocity  4  times 
as  great.  We  may  also  add  that  the  power  which  a  belt  of  given 
strength  can  transmit  varies  directly  as  the  speed  of  the  belt. 
For,  in  this  case,  P=Fv,  in  which  v  is  the  belt  speed,  and  F  is 
the  difference  in  tension  between  the  tight  and  the  slack  belt. 

PROBLEMS 

1.  A  certain  steel  bar  10  ft.  in  length  and  2  sq.  in.  in  cross  section  is 
elongated  0.22  in.  by  a  50-ton  pull.     What  is  Young's  modulus  E  for  this 
specimen? 

2.  A  steel  wire  3  meters  in  length  and  2  mm.  in  diameter  supports  a  load 
of  10  kilos.     How  much  will  the  wire  elongate  under  this  load,  if  Young's 
modulus  for  the  wire  is  1.9X1012  dynes  per  cm.2? 

3.  How  much  will  a  copper  wire  10  meters  in  length  and  2  sq.  mm.  in  cross 
section  stretch  under  a  load  of  3  kilos?     Young's  modulus  for  copper  is  1.2 
X1012  dynes  per  cm.2 

4.  A  certain  shaft  A  can  safely  transmit  50   H.P.     What  power  can  be 
transmitted  by  a  shaft  of  the  same  material    having  twice  as  great  a  di- 
ameter and  3  times  as  great  an  angular  velocity  as  A? 

6.  An  oak  timber  3  in.  by  12  in.  rests  edgewise  upon  two  supports  which 
are  8  ft.  apart.  How  much  will  the  beam  bend  (deflect  at  the  middle)  under 
a  load  of  1000  Ibs.  applied  midway  between  the  supports?  Young's  Modulus 
for  oak  is  1,500,000  Ibs.  per  sq.  in. 

6.  How  much  would  the  1000-lb.  load  bend  the  timber  (Prob.  5)  if  the 
timber  rested  flatwise  upon  the  supports? 


CHAPTER  X 
PROPERTIES  OF  LIQUIDS  AT  REST 

112.  Brief  Mention  of  Properties. — Some  of  the  properties 
of  liquids  in  addition  to  the  general  properties  of  matter  (Sec. 
101),  are  Viscosity,  Solvent  Action,  Diffusion,  Osmosis,  Pressure 
Production,  Pressure  Transmission,  and  Surface  Tension. 

Elasticity. — The  only  kind  of  elasticity  that  liquids  or  gases 
can  have  is  of  course  volume  elasticity  (Sec.  110).  Liquids  (also 
gases)  are  perfectly  elastic,  that  is,  however  much  a  liquid  is 
compressed,  upon  removing  the  pressure  the  liquid  expands  to 
exactly  its  former  volume.  There  is  no  such  thing  as  elastic 
fatigue  or  elastic  limit  for  liquids.  It  requires  very  high  pressure 
to  produce  appreciable  compression  of  a  liquid.  Thus  a  pres- 
sure of  100  Ibs.  per  sq.  in.  applied  to  a  volume  of  water  causes  a 
shrinkage  of  only  1  part  in  3000. 

Viscosity. — If  a  vessel  filled  with  syrup  has  a  small  hole  made 
near  the  bottom,  the  syrup  will  flow  slowly  through  the  hole. 
If  the  vessel  were  filled  with  water  instead,  it  would  be  found  that 
the  water,  having  less  viscosity,  would  flow  much  more  quickly 
through  the  hole.  Syrup  is  said  to  be  viscous,  and  water  mobile. 
Water,  however,  has  some  viscosity.  Glycerine  has  greater 
viscosity  than  water  but  less  than  molasses.  Viscosity  arises 
from  internal  friction,  that  is,  friction  between  the  molecules 
of  the  liquid.  The  greater  viscosity  of  glycerine  as  compared 
with  that  of  water  is  then  due  to  the  fact  that  glycerine  molecules 
do  not  glide  over  each  other  so  readily  as  do  water  molecules. 

It  may  easily  be  observed  that  the  water  on  the  surface  of  a 
river  moves  more  rapidly  than  that  near  the  bottom,  and  also 
that  the  water  near  the  center  of  the  stream  moves  more  rapidly 
than  that  near  the  shore.  This  difference  in  velocity  is  due  to 
friction  upon  the  bed  of  the  river  (and  upon  its  shores),  which 
causes  the  layers  very  near  the  bottom  to  move  very  slowly. 
These  slowly  moving  layers  of  water,  due  to  friction  of  water 
on  water,  i.e.,  due  to  the  viscosity  of  water,  tend  to  retard  the 
motion  of  the  layers  above.  The  greatest  retarding  effect  is 

155 


156  MECHANICS  AND  HEAT 

exerted  upon  the  nearest  layers,  and  the  least  upon  the  surface 
layer.  Hence  the  velocity  of  flow  gradually  increases  from  the 
bottom  up. 

Solvent  Action. — Some  solids  when  placed  in  certain  liquids 
slowly  disappear.  Thus  salt  readily  "dissolves"  in  water,  form- 
ing a  solution.  Paraffine  dissolves  in  kerosene,  but  not  in  water; 
while  salt  dissolves  in  water  but  not  in  kerosene.  When  water 
has  dissolved  all  of  the  salt  it  is  possible  for  it  to  hold  in  solution, 
the  brine  thus  formed  is  said  to  be  a  saturated  solution  of  salt. 
Solution  is  usually  attended  by  either  evolution  or  absorption 
of  heat;  i.e.,  by  either  heating  or  chilling  action. 

Gold,  zinc,  and  some  other  metals  dissolve  to  a  certain  extent 
in  mercury,  forming  gold  amalgam,  zinc  amalgam,  etc.  These 
amalgams  are  really  solutions  of  the  metals  in  mercury. 

Some  liquids  dissolve  in  other  liquids.  Thus,  if  some  ether 
and  water  are  thoroughly  stirred  together  in  a  vessel  and  then 
allowed  to  stand  a  moment,  the  water,  being  the  heavier,  settles 
to  the  bottom  and  the  layer  of  ether  rests  upon  it.  Upon  ex- 
amination it  will  be  found  that  there  is  about  10  per  cent,  ether 
in  the  water,  and  about  3  per  cent,  water  in  the  ether,  which 
shows  that  a  saturated  solution  of  ether  in  water  is  about  10  per 
cent,  ether,  while  a  saturated  solution  of  water  in  ether  is  about 
3  per  cent,  water. 

Some  liquids  dissolve  certain  gases.  Thus  water  dissolves 
air  to  a  slight  extent,  and  at  room  temperature  and  atmospheric 
pressure,  water  dissolves  450  times  its  volume  of  hydrochloric 
acid  gas  (HC1),  or  600  times  its  volume  of  ammonia  gas  (NH3). 
What  is  known  commercially  as  ammonia  or  as  hydrochloric  acid 
is  simply  an  aqueous  solution  of  the  one  or  the  other  of  these 
gases.  Pure  liquid  ammonia  is  used  in  ice  manufacture  (Sec. 
200).  Hydrochloric  acid  gas  can  be  condensed  to  a  liquid, 
thus  forming  pure  liquid  hydrochloric  acid,  by  subjecting  it  to 
very  high  pressure  and  low  temperature.  A  given  volume  of 
water  will  dissolve  about  an  equal  volume  of  carbon  dioxide 
(CO2)  at  ordinary  pressure  and  temperature.  Under  greater 
pressure  it  dissolves  considerably  more,  and  is  then  called  soda 
water.  When  drawn  from  the  fountain,  the  pressure  upon  it 
is  reduced,  and  the  escaping  CO2  produces  effervescence. 

Diffusion. — Many  liquids  if  placed  in  the  same  vessel,  mix 
even  though  of  quite  different  densities.  Thus,  if  some  ether  is 
very  carefully  introduced  onto  the  surface  of  some  water  in  such 


PROPERTIES  OF  LIQUIDS  AT  REST  157 

a  way  as  to  prevent  mixing  when  introducing  it,  it  will  be  found 
after  a  time  that  the  heavier  liquid  (water)  has  diffused  upward 
into  the  ether  until  the  latter  contains  about  3  per  cent,  water, 
while  the  ether,  although  lighter,  has  diffused  downward  into  the 
water. 

Osmosis. — Osmosis  is  the  mixing  or  diffusing  of  two  different 
liquids  or  gases  through  a  membrane  that  separates  them. 
Membranes  of  animal  or  plant  tissue  readily  permit  such  diffu- 
sion of  certain  substances  through  them.  Thus  a  bladder  filled 
with  water  does  not  leak,  but  if  lowered  into  a  vessel  of  alcohol 
it  slowly  collapses.  This  shows  that  the  water  passes  readily 
through  the  bladder;  the  alcohol  less  readily,  or  not  at  all. 
On  the  other  hand,  if  a  rubber  bag  is  filled  with  water  and  is 
then  lowered  into  a  vessel  of  alcohol,  it  becomes  more  and  more 
distended,  and  may  finally  burst.  In  this  case  it  is  the  alcohol 
which  passes  most  readily  through  the  separating  membrane. 

If  a  piece  of  parchment  or  other  such  membrane  is  tied  tightly 
across  the  mouth  of  an  inverted  funnel  filled  with  sugar  solution, 
and  the  funnel  is  placed  in  water,  it  will  be  observed  that  the 
solution  slowly  rises  in  the  stem.  By  prolonging  the  stem  a 
rise  of  several  feet  may  be  obtained.  Obviously  the  pure  water 
passes  more  readily  through  the  membrane  than  does  the 
sweetened  water,  or  sugar  solution.  If  the  solution  is  1 . 5  per 
cent,  sugar  (by  weight),  it  will  finally  rise  in  the  stem  about  34 
ft.  above  the  level  of  the  water  outside. 

Since  a  column  of  water  34  ft.  in  height  exerts  a  pressure  of 
about  one  atmosphere  (Sec.  136),  which  pressure  in  this  case  would 
tend  to  force  the  solution  through  the  membrane  into  the  water, 
it  follows,  when  equilibrium  is  reached,  i.e.,  when  no  further  rise 
of  the  column  occurs,  that  the  Osmotic  Pressure  developed  by  the 
tendency  of  the  water  to  pass  through  the  membrane  into  the  solu- 
tion, must  be  one  atmosphere  for  a  1 . 5  per  cent,  sugar  solution. 

With  weak  solutions,  the  osmotic  pressure  varies  approxi- 
mately as  the  strength  of  the  solution.  Thus  a  3  per  cent,  sugar 
solution  would  develop  an  osmotic  pressure  of  about  2  atmos- 
pheres. The  osmotic  pressure  also  differs  greatly  for  different 
solutions.  Thus,  for  example,  if  a  solution  of  common  salt  is 
used  the  osmotic  pressure  developed  will  be  much  more  than  for 
the  same  strength  (in  per  cent.)  of  sugar  solution. 

In  accordance  with  the  kinetic  theory  of  matter  (Sec.  99) 
we  may  explain  osmotic  pressure  by  assuming,  in  the  case  cited 


158  MECHANICS  AND  HEAT 

above,  that  the  water  molecules  in  their  vibratory  motion,  pass 
more  readily  through  the  animal  membrane  (the  bladder)  than 
do  the  more  complicated  and  presumably  larger  alcohol  mole- 
cules. This  is  the  commonly  accepted  explanation.  The  fact, 
however,  that  substituting  a  rubber  membrane  reverses  the  ac- 
tion, makes  it  seem  probable  that  something  akin  to  chemical 
affinity  between  the  membrane  and  the  liquids  plays  an  impor- 
tant role.  From  this  standpoint,  we  would  explain  this  reversal 
in  osmotic  action  by  stating  that  the  rubber  membrane  has 
greater  affinity  for  alcohol  than  for  water;  while  in  the  case  of 
animal  tissue  the  reverse  is  true.  Osmosis  plays  an  important 
part  in  the  physiological  processes  of  nutrition,  secretion  by 
glands,  etc.,  and  in  the  analogous  processes  in  plant  life.  Gases 
also  pass  in  the  same  way  through  membranes.  In  this  way 
the  blood  is  purified  in  the  capillary  blood-vessels  of  the  lungs 
by  the  oxygen  in  the  adjacent  air  cells  of  the  lungs. 

In  chemistry,  Dialysis,  the  process  by  which  crystalloids,  such 
as  sugar  and  salt  are  separated  from  the  colloids — starch,  gum, 
albumin,  etc.,  depends  upon  osmosis.  Crystalloids  pass  readily 
through  certain  membranes;  colloids,  very  slowly,  or  not  at  all. 
In  case  of  suspected  poisoning  by  arsenic  or  any  other  crystal- 
loid, the  contents  of  the  stomach  may  be  placed  on  parchment 
paper  floating  on  water.  In  a  short  time  the  crystalloids  (only) 
will  have  entered  the  water,  which  may  then  be  analyzed. 

Pressure  and  its  Transmission. — Liquids  exert  and  also  trans- 
mit pressure.  In  deep-sea  diving  the  pressure  sustained  by  the 
divers  is  enormous.  By  means  of  our  city  water  mains,  pressure 
is  transmitted  from  the  pumping  station  or  supply  tank  to  all 
parts  of  the  system.  (This  property  will  be  fully  discussed  in 
Sees.  113  and  114.  Surface  Tension  will  be  considered  in  Sec. 
124.) 

113.  Hydrostatic  Pressure. — The  study  of  fluids  at  rest  is 
known  as  Hydrostatics,  and  that  of  fluids  in  motion,  as  Hydraulics. 
From  their  connection  with  these  subjects  we  have  the  terms 
hydrostatic  pressure  and  hydraulic  machinery  such  as  hydraulic 
presses,  hydraulic  elevators,  etc. 

A  liquid,  because  of  its  weight,  exerts  a  force  upon  any  body 
immersed  in  it.  This  force,  divided  by  the  area  upon  which 
it  acts,  is  called  the  Hydrostatic  Pressure,  or 

TT    ,  total  force 

Hydrostatu  pressure  (average)  =  — - 


PROPERTIES  OF  LIQUIDS  AT  REST  159 

Note  that  pressure,  like  all  stresses  (Sec.  110),  is  the  total  force 
applied  divided  by  the  area  to  which  it  is  applied.  The  unit 
in  which  to  express  pressure  will  therefore  depend  upon  the 
units  in  which  the  force  and  the  area  are  expressed.  Some  units  of 
pressure  are  the  poundal  per  square  inch,  the  pound  per  square  inch, 
the  pound  per  square  foot,  and  the  dyne  per  square  centimeter. 
Let  it  be  required  to  find  the  pressure  at  a  depth  h  below  the 
surface  of  the  liquid  of  density  d  in  the  cylindrical  vessel  of 
radius  r,  Fig.  66.  The  formula  for  the  pres- 
sure on  the  bottom  of  the  vessel  is,  by  defi- 
nition, 

total  force  on  the  bottom 
Pressure  =       - 


The  force  on   the  bottom  is  obviously  the 

weight  W  of  the  liquid,  and  the  area  A  is 

Trr2;  so  that  the  pressure  is  W/irr2.     We  may 

express   W  in   dynes,   poundals,  or  pounds 

force,  and  Trr2  in  square  centimeters,  square 

inches,  etc.     The  weight   in   dynes   is  Mg,  pIG  gg 

but  the  mass  M  in  grams  is  the  product  of 

•jrr2h,  the  volume  of  the  liquid  in  cubic  centimeters,   and  d  its 

density  in  grams  per  cubic  centimeter.     Hence 

force     W     Mg     irrVidg        , 

Pressure  p  =  --  =  -r-  =  —  sr  =  -  ^    =  hag  dynes  per  cm.     (68) 
area      A       Trr2         Trr2 

In  the  British  system,  -irr2h  would  be  the  volume  of  the  liquid 
column  in  cubic  feet,  and  d  the  density  in  pounds  per  cubic  foot; 
so  that  irr^hd  would  be  the  weight  in  pounds,  and  irr^hdg  would 
be  the  weight  in  poundals.  Note  that  1  Ib.  =  g  poundals,  i.e., 
32.17  poundals  (Sec.  32).  Accordingly,  the  pressure  produced  by 
a  column  of  liquid  whose  height  is  h  feet  is  hdg  poundals  per 
square  foot,  or  hd  pounds  per  square  foot. 

114.  Transmission  of  Pressure.  —  If  a  tube  A  (Fig.  67)  with 
side  branches  B,  C,  D  and  E,  is  filled  with  water,  it  will  be 
found  that  the  water  stands  at  the  same  level  in  each  branch 
as  shown.  Further,  if  A  contains  four  small  holes,  a,  b,  c,  and 
d,  all  of  the  same  size  and  at  the  same  level,  and  covered  by  valves 
a',  b',  c',  and  d'}  respectively,  it  will  be  found  that  it  requires  the 
same  amount  of  force  to  hold  the  valve  a'  closed  against  the 
water  pressure  as  to  hold  &',  c',  or  d'  closed. 

If  the  branch  tube  B  were  removed,  everything  else  being  left 


160 


MECHANICS  AND  HEAT 


just  as  before,  it  is  evident  from  symmetry  that  a  small  valve 
at  e  in  order  to  prevent  water  from  coming  out  would  have  to 
resist  an  upward  pressure  at  e  (say  p%)  equal  to  the  upward 
pressure  at  c,  d,  etc.  With  B  in  place,  however,  the  water  does 
not  come  out  of  e,  but  is  at  rest;  hence  the  downward  pressure 
at  e  (say  pi)  due  to  the  column  of  water  in  B  must  just  balance 
the  above-mentioned  pressure  p2.  The  pressure,  pi,  however, 
is  equal  to  hdg  (Eq.  68).  If  the  pressure  at  a,  6,  c,  and  d,  is 
represented  by  pa,  pb,  pc,  and  Pd  respectively,  we  have 


The  experiment  shows,  then,  that  in  liquids  the  pressure  (a) 
is  exactly  equal  in  all  directions  at  a  given  point  (see  also  experi- 
ment below);  (6)  is  transmitted  undiminished  to  all  points  at  the 


FIG.  67. 

same  level;  and  (c)  is  numerically  hdg,  in  which  h  is  the  vertical 
distance  from  the  point  in  question  to  the  upper  free  surface  of 
the  liquid  causing  the  pressure. 

The  above  three  facts  or  principles  (a),  (&),  and  (c)  are  funda- 
mental to  the  subjects  of  hydrostatics  and  hydraulics.  They 
are  utilized  in  our  city  water  systems,  in  hydraulic  mining,  and 
in  all  hydraulic  machinery.  They  must  be  reckoned  with  in 
deep-sea  diving  and  in  the  construction  of  mill  dams  and  coffer- 
dams. In  these  and  hundreds  of  other  ways  these  principles 
find  application. 

The  greater  pressure  in  the  water  mains  in  the  low-lying  por- 
tions of  the  city  as  compared  with  the  hill  sections,  is  at  once  ex- 
plained by  (c),  noting  that  the  vertical  distance  from  these  points 
to  the  level  of  the  water  in  the  supply  tank  is  greater  for  these 
places  than  it  is  on  the  hills. 

An  exceedingly  simple  experimental  proof  of  the  principle 
(a)  may  be  arranged  as  follows:  A  glass  jar  containing  water 


PROPERTIES  OF  LIQUIDS  AT  REST 


161 


has  placed  in  it  several  glass  tubes  which  are  open  at  both  ends. 
Some  of  these  tubes  are  bent  more  or  less  at  the  lower  end,  so 
that  the  lower  opening  in  some  cases  faces  upward,  in  others 
downward,  and  still  others  horizontally  or  at  various  angles  of 
inclination.  If  these  openings  are  all  at  the  same  depth,  the 
fact  that  the  water  stands  at  the  same  height  in  all  of  the  tubes, 
that  is,  at  the  general  level  of  the  water  in  the  vessel,  shows  that 
the  outward  pressure  at  each  lower  opening  must  be  the  same. 
Consequently,  since  no  flow  takes  place,  the  inward  pressure  at 
each  opening,  which  is  due  to  the  general  pressure  of  the  main 
body  of  water,  and  which  is  exerted  in  various  directions  for  the 
different  tubes,  must  be  the  same  for  all. 

Pressure  Perpendicular  to  Walls. — The  pressure  exerted  by  a 
liquid,  against  the  wall  of  the  containing  vessel  at  any  point 
is  always  perpendicular  to  the  wall  at  that  point.  For  if  the 
pressure  were  aslant  with  reference  to  the  wall  at  any  point,  it 
would  have  a  component  parallel  to  the  wall  which  would  tend 
to  move  the  liquid  along  the  wall.  We  know,  however,  that  the 
liquid  is  at  rest;  hence  the  pressure  can  have  no  component 
parallel  to  the  wall,  and  is  therefore  perpen- 
dicular to  the  wall  at  all  points. 

115.  The  Hydrostatic  Paradox. — A  small 
body  of  liquid,   for  example  the  column  in 
tube  B  (Fig.  68),  may  balance  a  large  body 
of  liquid,    such    as  the   column  in  tube  A. 
This  is  known  as   the  Hydrostatic  Paradox. 
From  the  preceding  sections,  we  see  that  the 
pressure  tending  to  force  the  liquid  through  C 
in  the  direction  of  arrow  6,  is  hdg,  due  to  the 

column  of  liquid  B,  while  the  pressure  tending  to  force  it  in  the 
direction  of  arrow  a  is  likewise  hdg  due  to  the  column  of  liquid  A. 
Evidently  the  liquid  in  C  will  be  in  equilibrium  and  will  not  tend 
to  move  either  to  the  right  or  left  when  these  two  pressures  are 
equal,  i.e.,  when  h  is  the  same  for  both  columns.  Thus,  viewed 
from  the  pressure  standpoint,  we  see  that  there  is  nothing  para- 
doxical in  the  behavior  of  the  liquid.  If  A  contained  water  and 
B  contained  brine,  then  the  liquid  level  in  A  would  be  higher 
than  in  B  (Sec.  116). 

116.  Relative  Densities  of  Liquids  by  Balanced  Columns. — A 
very  convenient  method  of  comparing  the  densities  of  two  liquids, 
is  that  of  balanced  columns,  illustrated  in  Fig.  69.     A  U-shaped 


FIG-  68. 


162 


MECHANICS  AND  HEAT 


glass  tube,  with  arms  A  and  B,  contains  a  small  quantity  of, 
mercury  C,  as  shown.     If  water  is  poured  into  the  arm  A  and 
at  the  same  time  enough  of  some  other  liquid,  e.g.,  kerosene,  is 
poured  into  the  arm  B  to  just  balance  the  pressure  of  the  water 
column  A,  as  shown  by  the  fact  that  the  mercury  stands  at  the 
same  level  in  both  arms;  then  it  is  evident  that  the  pressure  p2 
due  to  the  kerosene,  which  tends  to  force  C  to  the  left,  must 
be  equal  to  the  pressure  p\  due  to  the  water,  which  tends  to  force 
C  to  the  right.     But  the  former  pressure  is  h2d2g 
while  the  latter  pressure  is  hidig,  in  which  hi  and 
h2  are  the  heights  of  the  water  and  the  kerosene 
columns  respectively,  and  dt  and  d2  the  respective 
densities  of  the  two  liquids.     Hence 

hidig  =  h2d2g 

d2     hi  hi 

or-r  =  T-,  or  d2  =  ^-dl 


FIG. 


The  density  d\  of  water  is  almost  exactly  1  gm. 
per  cm.3;  therefore  if  hi  is  found  to  be  40  cm.,  and 
h2  is  found  to  be  50  cm.,  then  the  density  of  kero- 
sene is  4/5  that  of  water  or  practically  0.8  gm.  per  cm.3 

117.  Buoyant  Force. — Any  body  immersed  in  a  liquid  experi- 
ences a  certain  buoyant  force.  This  force,  if  the  body  is  of 
small  density  compared  with  the  liquid,  causes  the  body  to  rise 
rapidly  to  the  surface.  Thus  cork  floats  on  water,  and  iron  on 
mercury.  This  buoyant  force  is  due  to  the  fact  that  the  upward 
pressure  on  the  body  is  greater  than  the  downward  pressure  on  it. 

Let  B,  Fig.  70,  be  a  cylindrical  body  immersed  in  a  vessel  of 
water.  Let  AI  and  A 2  be  the  areas  of  the  lower  and  upper  ends 
respectively,  and  let  pi  and  p2  be  the  corresponding  pressures. 
If  AI  is  3  times  as  far  below  the  surface  as  A2,  then  pi  will  equal 
3p2.  The  forces  on  the  sides  of  B  will  of  course  neutralize  each 
other  and  produce  neither  buoyant  nor  sinking  effect.  The 
entire  Buoyant  Force  of  the  water  uponZ?  is,  then,  Fi—F2,  in  which 
FI  is  the  upward  push  or  force  on  AI,  and  F2  the  much  smaller 
downward  push  on  A2.  Force,  however,  is  the  pressure  multi- 
plied by  the  area;  i.e., 


I,  and  F2  =  p2  A2,  or,  since 


Buoyant 


PROPERTIES  OF  LIQUIDS  AT  REST  163 

If  this  buoyant  force,  which  tends  to  make  the  body  rise,  is 
(a)  greater  than  the  weight  W  of  B,  which  of  course  tends  to 
make  it  sink,  the  body  will  move  upward — rapidly  if  much 
greater,  and  slowly  if  but  little  greater.  (6)  If  the  buoyant 
force  is  equal  to  W ,  then  B  will  remain  in  equilibrium  and  float 
about  in  the  liquid.  Finally  (c),  if  W  is  greater  than  the  buoy- 
ant force,  then  B  will  sink  to  the  bottom,  and  the  rapidity  with 
which  it  sinks  depends  upon  how  much  its  weight  exceeds  the 
buoyant  force. 

If  the  body  were  of  irregular  shape  such  as  C,  it  would  be 
very  difficult  to  find  its  area,  and  also  difficult  to  find  the  average 
vertical  components  of  pressure  on  the  upper 
and  lower  surfaces.  It  is,  nevertheless,  ob- 
vious that  the  average  downward  pressure  on 
the  body  would  be  less  than  the  average  up- 
ward pressure,  and  it  is  just  this  difference 
in  pressure  that  gives  rise  to  the  buoyant 
force  whatever  shape  the  body  may  have  (see 
Sec.  118).  The  horizontal  components  of 
pressure  would,  of  course,  have  no  tendency  J^Q  79. 

to  make  the  body  either  float  or  sink. 

118.  The  Principle  of  Archimedes. — If  any  body,  whatever 
be  its  shape,  e.g.,  A  (Fig.  71),  is  immersed  in  a  vessel  of  water,  it 
will  be  found  to  be  lighter  in  weight  than  if  it  were  weighed  in  air. 
This  difference  in  weight  is  referred  to  as  the  "Loss  of  Weight" 
in  water,  and  is  found  to  be  equal  to  the  weight  of  the  water  that 
would  occupy  the  space  now  occupied  by  A.  In  other  words, 
the  loss  of  weight  in  water  is  equal  to  the  weight  of  the  water  dis- 
placed. This  principle,  of  course,  holds  for  any  other  liquid, 
and  also  for  any  gas  (Sec.  134),  and  is  known  as  the  Principle 
of  Archimedes,  so  called  in  honor  of  the  Grecian  mathematician 
and  physicist  Archimedes  (B.  C.  287-212)  who  discovered  it. 

Theoretical  Proof  of  Archimedes'  Principle. — Imagine  the  body 
A  (Fig.  71)  to  be  replaced  by  a  body  of  water  A'  of  exactly  the 
same  size  and  shape  as  A  and  enclosed  in  a  membranous  sack 
of  negligible  weight.  It  is  evident  that  A'  would  have  no  tend- 
ency either  to  rise  or  to  sink.  It  then  appears  that  this  particular 
portion  of  water  loses  its  entire  weight,  hence  it  must  be  true  that 
the  buoyant  force  exerted  upon  A'  is  exactly  equal  to  its  weight. 
Since  this  buoyant  force  is  the  direct  result  of  the  greater  average 
pressure  upon  the  lower  side  than  upon  the  upper  side  of  the 


164 


MECHANICS  AND  HEAT 


body,  it  can  in  no  wise  depend  upon  the  material  of  which  the 
body  is  composed.  Consequently,  the  body  A  must  experience 
this  same  amount  of  buoyant  force,  and  therefore  must  lose  this 
same  amount  of  weight,  namely,  the  weight  of  the  water  displaced. 
Experimental  Proof  of  Archimedes'  Principle. — A  small  cylin- 
drical bucket  B  is  hung  from  the  beam  of  an  ordinary  beam 
balance,  and  a  solid  metal  cylinder  C  (Fig.  72)  which  accurately 
fits  and  completely  fills  the  bucket  is  suspended  from  it.  Suf- 
ficient mass  is  now  placed  in  the  pan  at  the  other  end  of  the  beam 
to  secure  a  "  balance. "  Next  a  large  beaker  of  water  is  so  placed 
that  the  solid  cylinder  is  immersed.  This,  of  course,  buoys  it 
up  somewhat  and  destroys  the  "balance."  Finally  the  bucket 
is  filled  with  water,  whereupon  it  will  be  found  that  exact  "bal- 


FIG.  71. 


FIG.  72. 


ance"  is  restored,  i.e.,  Fi=Fz.  This  fact  shows  that  the  weight 
of  the  water  in  the  bucket  just  compensates  for  the  buoyant 
force  that  arises  from  the  immersion  of  the  cylinder.  In  other 
words  the  loss  of  weight  experienced  by  the  cylinder  is  equal  to 
the  weight  of  the  water  which  fills  the  bucket,  and  is  therefore 
equal  to  the  weight  of  the  water  displaced  by  the  cylinder. 

119.  Immersed  Floating  Bodies. — In  case  the  body  A  (Fig. 
71)  is  denser  than  water,  it  will  weigh  more  than  the  water  which 
it  displaces  and  will  therefore  tend  to  sink.  If,  however,  it  has 
the  same  density  as  water,  the  buoyant  force  will  be  just  equal 
to  its  weight,  and  it  will  therefore  lose  its  entire  weight  and  float 
about  in  the  liquid. 

If  a  tall  glass  jar  is  about  one-third  filled  with  strong  brine 
and  is  then  carefully  filled  with  water,  the  two  liquids  will  mix 


PROPERTIES  OF  LIQUIDS  AT  REST  165 

slightly,  so  that  the  jar  will  contain  a  brine  varying  in  strength, 
and  hence  in  density,  from  that  which  is  almost  pure  water  at 
the  top,  to  a  strong  dense  brine  at  the  bottom.  If  pieces  of  resin, 
wax,  or  other  substances  which  sink  in  water  but  float  in  brine 
are  introduced,  they  will  sink  to  various  depths,  depending  upon 
their  densities.  Each  piece,  however,  sinks  until  the  buoyant 
force  exerted  upon  it  is  equal  to  its  weight,  that  is,  until  the 
weight  of  the  liquid  displaced  is  equal  to  its  own  weight. 

Occasionally  the  query  arises  as  to  whether  heavy  bodies  such 
as  metals  will  sink  to  the  bottom  of  the  ocean.  They  certainly 
do,  regardless  of  the  depth.  To  be  sure,  the  enormous  pressure 
at  a  great  depth  compresses  the  water  slightly,  making  it  more 
dense,  and  hence  more  buoyant.  The  increase  in  density  due 
to  this  cause,  however,  even  at  a  depth  of  one  mile  amounts 
to  less  than  1  per  cent,  (closely  3/4  per  cent).  Since  the  compres- 
sibility of  metals  is  about  1/100  as  great  as  that  of  water,  its 
effect  in  this  connection  may  be  ignored.  Substances,  however, 
which  are  more  readily  compressed  than  water,  e.g.,  porous  sub- 
stances containing  air,  actually  become  less  buoyant  at  great 
depths. 

120.  Application  of  Archimedes'  Principle  to  Bodies  Floating 
Upon  the  Surface. — If  a  piece  of  wood  that  is  lighter  than  water 
is  placed  in  water,  it  sinks  until  the  weight  of  the 
water   displaced    is  equal  to  its  own  weight.     If 
placed  in  brine  it  will  likewise  sink  until  the  weight 
of  the  liquid  displaced  is  equal  to  its  own  weight; 
but  it  will  not  then  sink  so  deep.    A  boat,  which  with 
its  cargo  weighs  1000  tons,  is  said  to  have  1000  tons 
"displacement,"  because  it  sinks  until  it  displaces 
1000  tons  of  water.     As  boats  pass  from  the  fresh        FIG.  73. 
water  into  the  open  sea  they  float  slightly  higher. 

If  a  wooden  block  B  (Fig.  73)  is  placed  in  water  and  comes  to 
equilibrium  with  the  portion  mnop  immersed,  then  the  volume 
mnop  is  the  volume  of  water  displaced,  and  the  weight  of  this 
volume  of  water  is  equal  to  the  entire  weight  of  the  block. 
Further,  if  d  is  9/10  c,  we  know  that  the  block  of  wood  displaces 
9/10  of  its  volume  of  water,  hence  its  density  is  0.9  gm.  per  cm.3 
(since  a  cm.3  of  water  weighs  almost  exactly  1  gm.). 

Ice  is  about  9/10  as  dense  as  sea  water;  consequently  icebergs 
float  with  approximately  9/10  of  their  volume  immersed  and  1/10 
above  the  surface.  If  some  projecting  points  are  100  ft.  above 


166  MECHANICS  AND  HEAT 

the  sea,  it  does  not  follow,  of  course,  that  the  iceberg  extends  900 
ft.  below  the  surface. 

121.  Center  of  Buoyancy.  —  If  a  rectangular  piece  of  wood  is  placed  in 
water  in  the  position  shown  at  the  left  in  Fig.  74,  the  center  of  gravity  of 
the  displaced  water  mnop  is  at  C.     This  point  C  is  called  the  Center  of 
Buoyancy.     It  is  the  point  at  which  the  entire  upward  lift  or  buoyant 
force  F,  due  to  the  water,  may  be  considered  as  concentrated.    The  center 
of  gravity,  marked  G,  is  the  point  at  which  the  entire  weight  W  of  the 
block  of  wood  may  be  considered  as  concentrated.     The  block  in  this 
position  is  unstable,  since  the  least  tipping  brings  into  play  a  torque  (as 

shown  at  the  right  in  Fig.  74)  tending  to  tip  it  still 
farther.  Consequently  the  block  tips  over  and 
floats  lengthwise  on  the  water.  For  the  same  rea- 
son logs  do  not  float  on  end,  but  lie  lengthwise 
on  the  water. 

If  a  sufficiently  large  piece  of  lead  were  fastened 
to  the  bottom  of  the  block  of  wood  so  as  to  bring 
FIG.  74.  its  center  of  gravity  below  its  center  of  buoyancy, 

the  block  would  then  be  stable  when  floating  on  end. 
Ballast  is  placed  deep  in  the  hold  of  a  vessel  in  order  to  lower  the  center 
of  gravity.  It  does  not  necessarily  follow,  however,  that  the  center  of 
gravity  of  ship  and  cargo  must  be  below  the  center  of  buoyancy  of 
the  ship.  For,  as  the  ship  rolls  to  the  right,  say,  the  form  of  the  hull  is 
such  that  the  center  of  buoyancy  shifts  to  the  right,  and  therefore  gives 
rise  to  a  righting  or  restoring  torque. 

122.  Specific  Gravity.—  The  Specific  Gravity  (S)  of  a  substance 
is  the  ratio  of  the  density  of  the  substance  to  the  density  of  water 
at  tne  same  temperature.     Representing  the  density  of  water 
by  d",  and  the  density  of  the  substance  referred  to  by  d,  we  have 

S  =  d/df  (69) 

Since  the  value  of  d'  is  very  nearly  one  (i.e.,  one  gm.  per  cm.3) 
at  ordinary  temperatures,  it  follows  that  the  Specific  Gravity 
of  a  substance  and  its  density  have  almost  the  same  value,  but 
they  must  not  be  considered  as  identical. 

Density,  however,  is  mass  divided  by  volume,  so  that  if  we 
consider  equal  volumes  of  the  substance  and  of  water,  and  repre- 
sent the  mass  of  the  former  by  M  and  that  of  the  latter  by  M'  , 
Eq.  69  may  be  written 


S  =  d/d'  =      ~  =  M/M'  =  Mg/M'g  =  W/W  (70) 


PROPERTIES  OF  LIQUIDS  AT  REST  167 

in  which  W  is  the  weight  of  a  certain  volume  of  the  substance 
and  W  the  weight  of  the  same  volume  of  water.  Hence  the 
specfic  gravity  of  a  substance  might  be  denned  as  the  ratio  of 
the  weight  of  a  certain  volume  of  that  substance  to  the  weight  of 
an  equal  volume  of  water. 

Specific  Gravity  of  a  Liquid. — If  a  bottle  full  of  liquid,  say 
kerosene,  weighs  Wi,  and  the  same  bottle  full  of  water  weighs 
W2,  while  the  empty  bottle  weighs  W3,  then  W\  —  W3  is  the  weight 
W  of  the  kerosene  in  the  bottle,  and  Wz  —  Ws  is  the  weight  W 
of  an  equal  volume  of  water;  hence  from  Eq.  70  we  have  for  the 
specific  gravity  of  kerosene 

W  _Wi_-TF3 
-W'~W2-WS 

If  a  piece  of  metal  which  has  first  been  weighed  in  air,  is  then 
immersed  in  water  and  again  weighed,  it  will  be  found  to  be 
lighter.  This  "loss  of  weight"  in  water,  i.e.,  its  weight  in  air 
minus  its  weight  in  water,  is  of  course  due  to  the  buoyant  force 
and  is  equal  to  the  weight  of  the  water  displaced.  If  the  piece 
of  metal  is  again  weighed  while  immersed  in  brine,  the  loss  of 
weight  will  be  equal  to  the  weight  of  the  brine  displaced.  This 
loss  of  weight  will  be  greater  than  the  former  loss.  Dividing 
it  by  the  former  loss  we  obtain  the  specific  gravity  of  the  brine. 

Specific  Gravity  of  a  Solid. — Evidently  the  volume  of  any  body 
immersed  in  water  is  exactly  equal  to  the  volume  of  water  which 
it  displaces.  Consequently  its  specific  gravity  is  the  ratio  of  the 
weights  of  these  two  volumes,  or  the  weight  of  the  body  in  air 
divided  by  its  loss  of  weight  in  water.  This  is  a  convenient 
method  for  determining  the  specific  gravity  of  irregular  solids, 
such  as  pieces  of  ore. 

If  a  stone  weighs  30  gm.  in  air  and  20  gm.  in  water,  then  the 
weight  of  the  water  it  displaces  must  be  10  gm. ;  so  that  the  stone 
weighs  3  times  as  much  as  the  same  volume  of  water  and  its 
specific  gravity  is,  therefore,  3.  Since  the  density  of  water  d' 
(Eq.  69)  is  very  slightly  less  than  1.0  at  room  temperature,  the 
density  d  of  the  stone  would  be  very  slightly  less  than  its  specific 
gravity. 

123.  The  Hydrometer. — The  hydrometer,  of  which  there  are 
several  kinds,  affords  a  very  rapid  means  of  finding  the  specific 
gravity  of  a  liquid.  It  is  also  sufficiently  accurate  for  most 
purposes.  The  most  common  kind  of  hydrometer  consists  of  a 


168 


MECHANICS  AND  HEAT 


glass  tube  A  (Fig.  75),  having  at  its  lower  end  a  bulb  B  contain- 
ing just  enough  mercury  or  fine  shot  to  properly  ballast  it  when 
floating.  From  Sec.  120  we  see  that  such  an  instrument  will 
sink  until  it  displaces  an  amount  of  water  equal  to  its  own  weight. 
To  do  this  it  will  need  to  sink  deeper  in  a  light  liquid  than  in  a 
heavy  liquid;  hence  the  depth  to  which  it  sinks  indicates  the 
specific  gravity  of  the  liquid  in  which  it  is  placed.  From  a 
scale  properly  engraved  upon  the  stem  of  the  hydrometer,  the 
specific  gravity  of  the  liquid  in  which  it  is  floating  may  be  read 
by  observing  the  mark  that  is  just  at  the  surface. 
Thus,  if  the  hydrometer  sinks  to  the  point  a  in  a  given 
liquid,  we  know  that  the  specific  gravity  of  the  liquid 
is  1.12,  i.e.,  it  is  1.12  times  as  dense  as  water.  The 
scale  shown  is  called  a  Specific  Gravity  Scale,  because 
the  specific  gravity  of  the  liquid  is  given  directly.  It 
will  be  observed  that  it  is  not  a  scale  of  equal  divisions. 
The  Beaume  Scale. — The  Beaume  Scale,  which  is 
very  much  used,  has  on  the  one  hand  the  advantage 
of  having  equal  scale  divisions;  but  on  the  other  hand 
it  has  the  disadvantage  that  it  is  entirely  arbitrary, 
and  that  its  readings  do  not  give  directly  the  specific 
gravity  of  the  liquid.  There  are  two  Beaume  scales, 
one  for  liquids  heavier  than  water,  the  other  for  liq- 
uids lighter  than  water. 

To  calibrate  a  hydrometer  for  heavy  liquids  it  is 
placed  in  water,  and  the  point  to  which  it  sinks  is 
marked  0°.  It  is  next  placed  in  a  15  per  cent,  brine 
(15  parts  salt  and  85  parts  water,  by  weight)  and  the 
point  to  which  it  sinks  is  marked  15°.  The  space  be- 
tween these  two  marks  is  then  divided  into  15  equal  spaces  and 
the  graduation  is  continued  down  the  stem.  If,  when  placed 
in  a  certain  liquid,  the  hydrometer  sinks  to  mark  20,  the  spe- 
cific gravity  of  the  liquid  is  20°  Beaume  heavy. 

For  use  in  light  liquids,  the  point  to  which  the  instrument  sinks  in  a 
10  per  cent,  brine  is  marked  0°,  and  the  point  to  which  it  sinks  in  water 
is  marked  10°.  The  space  between  these  two  marks  is  divided  into 
10  equal  spaces,  and  the  graduation  is  extended  up  the  stem.  If,  when 
placed  in  a  certain  liquid,  the  hydrometer  sinks  to  mark  14,  the  specific 
gravity  of  the  liquid  is  14°  Beaume  light. 

124.  Surface  Tension. — Small  drops  of  water  on  a  dusty  or 
oily  surface  assume  a  nearly  spherical  shape.  Small  drops  of 


FIG.  75. 


PROPERTIES  OF  LIQUIDS  AT  REST  169 

mercury  upon  most  surfaces  behave  in  the  same  manner.  Dew- 
drops  and  falling  raindrops  are  likewise  spherical.  When  the 
broken  end  of  a  glass  rod  having  a  jagged  fracture  is  heated  until 
soft,  it  becomes  smoothly  rounded.  These  and  many  other 
similar  phenomena  are  due  to  what  is  called  Surface  Tension 
(denned  in  Sec.  126). 

Surface  tension  arises  from  the  intermolecular  attraction  ^or 
cohesion)  between  adjacent  molecules.  Some  of  the  effects  of 
this  attraction  have  already  been  discussed  in  Sec.  102.  Certain 
experiments  indicate  that  these  molecular  forces  do  not  act  ap- 
preciably through  distances  greater  than  about  two-millionths 
of  an  inch.  A  sphere,  then,  of  two-millionths  inch  in  radius 
described  about  a  molecule  may  be  called  its  sphere  of  influence, 
or  sphere  of  molecular  attraction. 

Let  A,  B,  and  C  (Fig.  76)  represent  respectively  a  molecule  of 
water  well  below  the  surface,  one  very  near  the  surface,  and  one 


FIG.  76.  FIG.  77. 

on  the  surface;  and  let  the  circles  represent  their  respective 
spheres  of  molecular  action.  Evidently  A,  which  is  completely 
surrounded  by  water  molecules,  will  be  urged  equally  in  all 
directions  and  hence  will  have  no  tendency  to  move.  It  will 
therefore,  barring  friction,  not  require  any  force  to  move  it  about 
in  the  liquid;  but,  as  we  shall  presently  see,  it  will  require  a 
force  to  move  it  to  the  surface.  Accordingly,  work  is  done  in 
increasing  the  amount  of  surface  of  a  liquid  (Sec.  126),  e.g.,  as 
in  inflating  a  soap  bubble.  Part  of  B's  sphere  of  molecular 
attraction  projects  above  the  surface  into  a  region  where  there 
are  no  water  molecules,  and  hence  the  aggregate  downward  pull 
on  B  exerted  by  the  surrounding  molecules  is  greater  than  the 
upward  pull  upon  it.  In  the  case  represented  by  C,  there  is  no 
upward  pull,  except  the  negligible  pull  due  to  the  adjacent  mole- 
cules of  air.  Consequently  Bt  and  C,  and  all  other  molecules 
on  or  very  near  the  surface,  are  acted  upon  by  downward  (inward) 


170  MECHANICS  AND  HEAT 

forces.  The  nearer  a  molecule  approaches  to  the  surface,  the 
greater  this  force  becomes. 

In  Fig.  77,  A  represents  a  small  water  drop  and  a,  b,  c,  d,  etc., 
surface  molecules.  Since  every  surface  molecule  tends  to  move 
inward,  the  result  is  quite  similar  to  uniform  hydrostatic  pressure 
on  the  entire  surface  of  the  drop.  But  such  pressure  would 
arise  if  the  surface  layer  of  molecules  were  a  stretched  mem- 
branous sack  (e.g.,  of  exceedingly  thin  rubber)  enveloping  the 
drop.  This  fact,  that  the  surface  layer  of  molecules  of  any 
liquid  behaves  like  a  stretched  membrane,  i.e.,  like  a  membrane 
under  tension,  makes  the  name  Surface  Tension  very  appro- 
priate. Although  there  is  no  stretched  film  over  the  drop,  the 
surface  molecules  differing  in  no  sense  from  the  inner  molecules 
except  that  they  are  on  the  surface,  it  is,  nevertheless,  very  conven- 
ient to  regard  the  phenomenon  of  surface  tension  as  arising  from 
the  action  of  stretched  films,  and  in  the  further  discussion  it  will 
be  so  regarded.  It  must  be  kept  in  mind,  however,  that  this 
is  merely  a  matter  of  convenience,  and  that  the  true  cause  of  sur- 
face tension  is  the  unbalanced  molecular  attraction  just  discussed. 

When  certain  insects  walk  upon  the  water,  it  is  easily  observed 
that  this  "membrane"  or  ''film"  sags  beneath  their  weight. 
A  needle,  especially  if  slightly  oily,  will  float  if  carefully  placed 
upon  water.  We  may  note  in  passing  that  the  weight  of  the 
water  displaced  by  the  sagging  of  the  surface  film  is  equal  to 
the  weight  of  the  needle  (Archimedes'  Principle). 

125.  Surface  a  Minimum. — Evidently  a  stretched  film  enclosing 
a  drop  of  liquid  would  cause  the  drop  to  assume  a  form  having 
the  least  surface,  i.e.,  requiring  the  least 
area  of  film  to  envelop  it.     The  sphere 
has  less  surface  for  a  given  volume  than 
JTIG  yg  any  other  form  of  surface.     Hence  drops 

of  water  are  spherical.     For  the  same 

reason  soap  bubbles,  which  are  merely  films  of  soapy  water  en- 
closing air,  tend  to  be  spherical.  A  large  drop  of  water,  or 
mercury,  or  any  other  liquid  is  not  spherical  if  resting  upon  a 
surface,  but  is  flattened,  due  to  its  weight  (see  Fig.  78).  Quite 
analogous  to  this  is  the  fact  that  a  small  rubber  ball  filled  with 
water  and  resting  upon  a  plane  surface  will  remain  almost  spher- 
ical; while  a  large  ball  made  of  equally  thin  rubber  would  flatten 
quite  appreciably,  due  to  its  greater  weight. 

If  the  effect  of  the  weight  of  the  drop  is  removed,  this  flatten- 


PROPERTIES  OF  LIQUIDS  AT  REST  171 

ing  does  not  take  place  even  for  very  large  drops.  Thus,  if  a 
mixture  of  alcohol  and  water  having  the  same  density  as  olive 
oil  is  prepared,  it  will  be  found  that  a  considerable  quantity  of 
this  oil  retains  the  spherical  form  when  carefully  introduced  well 
below  the  surface  of  the  mixture. 

That  a  film  tends  to  contract  so  as  to  have  a  minimum  area, 
and  that  in  so  doing  it  exerts  a  force,  is  beautifully  illustrated 
by  the  following  experiment.  If  the  wire  loop  B  (Fig.  79),  to 
which  is  attached  a  small  loop  of  thread  a,  is  dipped  into  a  soap 
solution  and  withdrawn,  it  will  have  stretched  across  it  a  film 
in  which  the  loop  a  "floats"  loosely  as  indicated.  Evidently 
the  film,  pulling  equally  in  all 
directions  on  a,  has  no  tendency 
to  stretch  it.  If,  however,  the 
film  within  a  is  broken,  the  in- 
ward pull  disappears,  whereupon 
the  outward  pull  causes  the  loop  Fi^  79 

to    assume    the    circular    form 

shown  at  the  right  (Fig.  79).  A  loop  has  its  maximum  area  when 
circular;  consequently,  the  annular  film  between  the  thread  and 
the  wire  must  have  a  minimum  area  when  the  thread  loop  is 
circular. 

If  a  piece  of  sealing  wax  with  sharp  corners  is  heated  until 
slightly  plastic,  the  corners  are  rounded,  due  to  surface  tension 
of  the  wax;  and  in  this  rounding  process  the  amount  of  surface  is 
reduced.  Glass  and  all  metals  behave  in  the  same  way  when 
sufficiently  heated.  All  metals  when  melted,  indeed  all  sub- 
stances when  in  the  liquid  state,  exhibit  surface  tension.  This 
property  is  utilized  in  making  fine  shot  by  dropping  molten  lead 
through  the  air  from  the  shot  tower.  During  the  fall,  the  drops 
of  molten  lead  cool  in  the  spherical  form  produced  by  surface 
tension. 

126.  Numerical  Value  of  Surface  Tension. — The  Surface 
Tension  T  of  a  liquid  is  numerically  the  force  in  dynes  with  which 
a  surface  layer  of  this  liquid  one  centimeter  in  width  resists  being 
stretched.  There  are  several  methods  of  finding  the  surface 
tension,  in  all  of  which  the  force  required  to  stretch  a  certain 
width  of  surface  layer  is  first  determined.  This  force,  divided 
by  the  width  of  the  surface  layer  stretched,  gives  the  value  of 
the  surface  tension. 

The  simplest  method  of  finding  the  surface  tension  is  the  follow- 


172  MECHANICS  AND  HEAT 

ing:  An  inverted  U  of  fine  wire  1/2  cm.  in  width  is  immersed 
in  a  soap  solution  (Fig.  80)  and  then  suspended  from  a  sensitive 
Jolly  balance.  (The  Jolly  balance  is  practically  a  very  sensitive 
spring  balance.)  Since  the  film  across  the  U  has  two  surfaces, 
one  toward  and  one  away  from  the  reader,  it  is  evident  that  in 
raising  the  U,  a  surface  layer  1  cm.  in  width  must  be  stretched. 
Hence  the  reading  of  the  Jolly  balance  (in  dynes)  immediately 
before  the  film  breaks  minus  the  reading  after,  gives  the  surface 
tension  for  the  soap  film  in  dynes  per  centimeter.  For  pure 
water,  T  is  approximately  80  dynes  per  cm.  Its  value  decreases 
due  to  rise  in  temperature,  and  also  due  to  the  presence  of  im- 
purities (Sec.  127.)  Observe  that  T  is  numerically  the  force  re- 
quired to  keep  stretched  a  surface  layer  having 
a  width  (counting  both  sides)  of  1  cm. 

In  raising  the  wire  (Fig.  80)  a  distance  of  1 
cm.,  a  force  of  80  dynes  (for  pure  water)  must 
be  exerted  through  a  distance  of  1  cm.,  that  is, 
80  ergs  of  work  must  be  done.  But  1  cm.2  of 
surface  has  been  formed;  showing  that  80  ergs  of 
work  are  required  to  form  1  cm.2  of  surface.  In 


FIG  80  other  words,  80  ergs  of  work  are -required  to 

cause  enough  molecules  to  move  from  position 
A  to  that  of  C  (Fig.  76)  to  form  1  cm.2  of  additional  surface. 

Observe  that  a  soap  bubble  has  an  outer  and  an  inner  surface. 
Between  these  two  surfaces  is  an  exceedingly  thin  layer  of  soapy 
water.  This  soapy  water,  as  it  flows  down  between  the  two 
surfaces,  forms  the  drop  which  hangs  below  the  bubble  and  at 
the  same  time  causes  other  portions  of  the  bubble  to  become 
thinner  and  thinner  until  it  finally  bursts.  The  greater  vis- 
cosity of  soapy  water,  as  compared  with  pure  water,  causes  the 
downward  flowing  to  be  much  slower  than  with  pure  water,  and 
therefore  causes  a  soap  bubble  to  last  much  longer  than  a  water 
bubble. 

In  blowing  a  soap  bubble,  work  is  done  upon  the  film  in  in- 
creasing its  area;  on  the  other  hand,  if  the  film  is  permitted  to 
contract  by  forcing  air  out  through  the  pipestem,  work  is  evi- 
dently done  by  the  film.  Barring  friction,  these  two  amounts 
of  work  must  be  equal. 

In  another  method  of  determining  surface  tension,  quite  similar 
in  principle  to  the  one  just  given,  a  wire  ring  suspended  in  a  hori- 
zontal position  from  a  Jolly  balance  is  lowered  until  it  rests  flat 


PROPERTIES  OF  LIQUIDS  AT  REST  173 

upon  the  water,  and  is  then  raised,  say  1/16  inch.  In  this  posi- 
tion it  would  be  found  that  a  film  tube  of  water,  having  the  di- 
ameter of  the  ring  and  a  length  of  1/16  inch,  connects  the  ring 
with  the  water  and  exerts  upon  the  ring  a  downward  pull.  The 
reading  of  the  Jolly  balance  just  before  this  film  breaks,  minus 
the  reading  after  (or  F\,  say),  gives  this  downward  pull.  The 
width  of  surface  layer  that  is  stretched  is  twice  the  circumference 
of  the  ring,  or  4irr.  Note  that  a  tube  has  an  outer  and  an  inner 
surface.  Hence 


which  may  be  solved  for  T. 

127.  Effect  of  Impurities  on  Surface  Tension  of  Water.  — 
Most  substances  when  dissolved  in  water  produce  a  marked 
decrease  in  its  surface  tension.     For  this  reason,  parings  of  cam- 
phor move  rapidly  over  the  surface  of  water  if 

dropped  upon  it.    Let  A,  B,  and  C,  Fig.  81,  be 

three  pieces   of    camphor  upon  the  surface  of 

water.     The  piece  A  dissolves  more  rapidly  from 

the  point  a  than  elsewhere,  so  that  the  surface 

tension  on  the  end  a  is  reduced  more  than  on  the 

opposite  end,  and  the  piece  moves  in  the  direc         FIG.  81. 

tion  of  the  stronger  pull,  as  indicated  by  the  ar- 

row.    In  the  case  of  C,  this  same  effect  at  c  gives  rise  to  a  rotary 

motion,  as  shown;  while  B  describes  a  curved  path  due  to  the 

same  cause. 

128.  Capillarity.  —  If  a  glass  A,  Fig.  82,  contains  water,  and 
another  glass  B  contains  mercury,  it  may  easily  be  observed  that 
most  of  the  surface  of  each  is  perfectly  flat,  as  shown,  but  that 
near  the  edge  of  the  glass,  the  water  surface  curves  upward,  while 
the  mercury  surface  curves  downward.     If  the  glass  A  were  made 
slightly  oily,  the  water  would  curve  downward;  while  if  B  were 
replaced  by  an  amalgamated  zinc  cup,  the  downward  curvature 
of  the  mercury  would  disappear.     Thus  the  form  of  the  surface 
depends  upon  both  the  liquid  and  the  containing  vessel. 

If  a  clean  glass  rod  is  dipped  into  water  and  then  withdrawn,  it 
is  wet.  This  shows  that  the  adhesion  between  glass  and  water 
exceeds  the  cohesion  between  the  water  particles.  For  the  water 
that  wets  the  glass  rod  must  have  been  more  strongly  attracted 
by  the  glass  than  by  the  rest  of  the  water,  or  it  would  not  have 
come  away  with  the  rod.  If  the  glass  rod  is  slightly  oily  it  will 


174 


MECHANICS  AND  HEAT 


FIG.  82. 


not  be  wet  after  dipping  it  into  the  water.  If  a  clean  glass  rod 
is  dipped  into  mercury  and  then  withdrawn,  the  fact  that  no 
mercury  comes  with  it  shows  that  the  cohesion  between  mercury 
molecules  exceeds  the  adhesion  between  mercury  and  glass  mole- 
cules. It  is,  indeed,  the  relative  values  of  cohesion  and  adhesion 
that  determine  surface  curvature  at  edges.  If  the  cohesion  of 
the  liquid  molecules  for  each  other  just  equals  their  adhesion  for 
the  substance  of  which  the  containing  vessel  is  made,  the  sur- 
face will  be  flat  from  edge  to  edge.  If  greater,  the  curvature  is 
downward  (B,  Fig.  82),  while  if  smaller,  it  is 
upward  (A,  Fig.  82).  Thus,  in  the  latter 
case,  the  water  at  the  edge  rises  above  the 
general  level,  wetting  the  surface  of  the 
glass,  simply  because  glass  molecules  at- 
tract water  molecules  more  strongly  than 

other  water  molecules  do.  This  phenomenon  is  most  marked 
in  the  case  of  small  tubes  (capillary  tubes)  and  is  therefore 
called  capillarity. 

129.  Capillary  Rise  in  Tubes,  Wicks,  and  Soil. — If  clean  glass 
tubes  a  and  b  (Fig.  83)  are  placed  in  the  vessel  of  water  A,  and 
c  and  d  in  the  vessel  of  mercury  B,  it  will  be  found  that  the  capil- 
lary rise  in  a  and  6,  and  the  capillary  depression  in  c  and  d  is 
greater  for  the  tube  of  smaller  bore.  Indeed,  it  will  be  shown  in 
the  next  section,  and  it  is  eas- 
ily observed  experimentally 
with  tubes  of  different  bore, 
that  a  given  liquid  rises  n  times 
as  high  in  a  tube  of  1/n  times 
as  large  bore. 

Any  porous  material  produces 
a  marked  capillary  rise  with  any 
liquid  that  wets  it.  There  are  FIG.  83. 

numerous   phenomena   due   to 

capillary  action,  many  of  which  are  of  the  greatest  importance. 
If  one  corner  of  a  lump  of  sugar,  or  clod  of  earth,  touches  the  water 
surface,  the  entire  lump  or  clod  becomes  moist.  Due  to  capil- 
larity, the  wick  of  a  lamp  carries  the  oil  to  the  flame  where  it  is 
burned.  If  the  substratum  soil  is  moist,  this  moisture,  during  a 
dry  time,  is  continually  being  carried  upward  to  the  roots  of  plants 
by  the  capillary  action  of  the  soil.  Capillarity  is  probably  an 
important  factor,  in.  connection  with  osmosis  (Sec.  112),  in  the 


PROPERTIES  OF  LIQUIDS  AT  REST  175 

transference  of  liquid  plant  food  from  the  rootlets  to  the  topmost 
parts  of  plants  and  trees. 

Cultivating  the  soil  to  the  depth  of  a,  few  inches  greatly  reduces 
the  amount  of  evaporation,  and  hence  helps  retain  the  moisture 
for  the  use  of  the  plants.  For,  stirring  the  ground  destroys,  in  a 
large  measure,  the  continuity,  and  hence  the  capillary  action, 
between  the  surface  soil  and  the  moist  earth  a  few  inches  below. 
Consequently  the  surface  soil  dries  more  quickly,  and  the  lower 
soil  more  slowly,  than  if  the  ground  had  not  been  stirred. 

130.  Determination  of  Surface  Tension  from  Capillary  Rise 
in  Tubes.  —  In  Fig.  84,  B  represents  a  capillary  tube  having  a 
bore  of  radius  r  cm.,  and  giving,  when  placed 
in  water,  a  capillary  rise  of  h  cm.  It  may  be 
considered  to  be  the  upward  putt  of  the  sur- 
face  layer  /  that  holds  the  column  of  water 
in  the  capillary  tube  above  the  level  of  the 
water  in  the  vessel.  The  weight  of  this  col- 
umn is  Trr'hdg  (see  Eq.  68,  Sec.  113).  The 
hemispherical  surface  layer  that  sustains  this 
weight,  however,  is  attached  to  the  bore  of  FIG.  84. 

the  tube  by  its  margin  abc  (as  shown  at  A),  so 
that  the  "width  of  surface"  (see  Sec.  126)  that  must  support  this 
weight  is  2irr,  consequently 


T  =  \rhdg  dynes  per  cm.  (71) 

The  above  method  is  the  one  most  frequently  used  for  deter- 
mining surface  tension.  It  is  usually  necessary  first  to  clean  the 
tube  with  nitric  acid  or  caustic  soda,  or  both,  and  then  carefully 
rinse  before  making  the  test. 

PROBLEMS 

1.  What  is  the  pressure  at  a  depth  of  2  mi.  in  the  ocean? 

2.  A  water  tank  has  on  one  side  a  hole  10  cm.  in  diameter.     What  force 
will  be  required  to  hold  a  stopper  in  the  hole  if  the  upper  edge  of  the  hole  is 
4  meters  below  the  water  level? 

3.  What  horizontal  force  will  a  lock  gate  40  ft.  in  width  exert  on  its  sup- 
ports if  the  depth  of  the  water  is  18  ft.  above  the  gate,  and  6  ft.  below  it? 

4.  Express  a  pressure  of  15  Ibs.  per  in.2  in  dynes  per  cm.2 

5.  The  right  arm  of  a  U-tube,  such  as  shown  in  Fig.  69,  contains  mercury 
only  and  the  left  arm  some  mercury  upon  which  rests  a  column  of  brine  60 


176  MECHANICS  AND  HEAT 

cm.  in  height.     The  mercury  stands  5.2  cm.  higher  in  the  right  arm  than 
in  the  left.     What  is  the  density  of  the  brine?     Sketch  first. 

6.  The  weight  of  a  stone  in  air  is  60  gm.,  in  water  38  gm.,  and  in  a  certain 
oil  42  gm.     What  is  the  sp.  gr.  (a)  of  the  stone?     (6)  of  the  oil? 

7.  Two  tons  increase  in  cargo  makes  a  boat  sink  1.2  in.  deeper  (in  fresh 
water).     What  is  the  area  of  a  horizontal  section  of  the  boat  at  the  water 
line? 

8.  A  marble  slab  (density  2 . 7  gm.  per  cm.3)  weighs  340  Ibs.  when  immersed 
in  fresh  water.     What  is  its  volume? 

9.  How  much  lead  must  be  attached  to  20  gm.  of  cork  to  sink  it  in  fresh 
water?     Consult  table  of  densities,  Sec.  101. 

10.  What  capillary  rise  should  water  give  in  a  tube  of  (a)  1  mm.  bore, 
(b)  2  mm.  bore? 

11.  A  wire  ring  of  5  cm.  radius  is  rested  flat  on  a  water  surface  and  is  then 
raised.     The  pull  required  to  raise  it  is  5  gm.  more  before  the  "film"  breaks 
than  it  is  after.     What  value  does  this  give  for  the  surface  tension? 


CHAPTER  XI 
PROPERTIES  OF  GASES  AT  REST 

131.  Brief  Mention  of  Properties. — Gases  have  all  the  prop- 
erties of  liquids  that  are  mentioned  in  Sec.  112  (to  which  section 
the  reader  is  referred)  except  solvent  action  and  surface  tension. 
Gases  have  also  properties  not  possessed  by  liquids,  one  of  which 
is  Expansibility. 

Viscosity. — The  viscosity  of  gases  is  much  smaller  than  that  of 
liquids,  but  it  is  not  zero,  nor  is  it  even  negligible.  In  order  to 
force  water  to  flow  rapidly  through  a  long  level  pipe,  the  pressure 
upon  the  water  as  it  enters  the  pipe  must  be  considerably  greater 
than  the  pressure  upon  it  as  it  flows  from  the  pipe.  This  differ- 
ence in  pressure  is  known  as  Friction  Head.  It  requires  a  pressure 
difference  or  pressure  drop  to  force  water  through  a  level  pipe 
because  of  the  viscosity  of  water.  To  produce  the  same  rate  of 
flow  through  a  given  pipe  would  require  a  much  greater  pressure 
drop  if  the  fluid  used  were  molasses  instead  of  water,  and  very 
much  smaller  drop  if  the  fluid  used  were  a  gas.  This  difference  is 
due  to  the  fact  that  the  viscosity  of  water  is  less  than  that  of 
molasses  and  greater  than  that  of  the  gas.  The  slight  viscosity 
of  illuminating  gas  necessitates  a  certain  pressure  drop  to  force 
the  required  flow  through  the  city  gas  mains. 

Usually  in  ascending  a  high  tower  there  is  a  noticeable,  steady 
increase  in  the  velocity  of  the  wind;  which  shows  that  the  higher 
layers  of  air  are  moving  more  rapidly  than  those  below  (compare 
with  the  flowing  of  a  river,  Sec.  112).  Indeed,  just  as  in  the  case 
of  the  layers  of  water  in  the  river,  each  layer  experiences  a  forward 
drag  due  to  the  layer  above  it  and  a  backward  drag  due  to  the 
layer  below  it,  and  therefore  moves  with  an  intermediate  velocity. 
The  lower  layers  are  retarded  by  trees  and  other  obstructions. 

It  is  probable  that  the  viscosity  of  gases  should  not  be  attrib- 
uted to  molecular  friction  but  rather  to  molecular  vibration  (see 
Kinetic  Theory  of  Matter,  Sec.  99).  Consider  a  rapidly  moving 
stratum  of  air  gliding  past  a  slower  moving  stratum  below.  As 
molecules  from  the  upper  stratum,  due  to  their  vibratory  motion, 

177 


178  MECHANICS  AND  HEAT 

wander  into  the  lower  stratum,  they  will,  in  general,  accelerate 
it;  whereas  molecules  passing  from  the  lower  stratum  to  the  upper 
will,  in  general,  retard  the  latter.  Thus,  any  interchange  of 
molecules  between  the  two  strata  results  in  an  equalization  of  the 
velocities  of  the  portions  of  the  strata  near  their  surface  of  separa- 
tion. Of  course  sliding  (molecular)  friction  would  produce  this 
same  result,  but  the  fact  that  a  rise  in  temperature  causes  the 
viscosity  to  decrease  in  liquids  and  increase  in  gases,  points  to  a 
difference  in  its  origin  in  the  two  cases.  As  a  gas  is  heated,  the 
vibrations  of  its  molecules,  according  to  the  Kinetic  Theory  of 
Gases  (Sec.  171),  become  more  violent,  thus  augmenting  the  above 
molecular  interchange  between  the  two  layers  and  thereby 
increasing  the  apparent  friction  between  them. 

Diffusion. — Diffusion  is  very  much  more  rapid  in  the  case  of 
gases  than  with  liquids,  probably(  because  of  greater  freedom  of 
molecular  vibration.  Thus  if  some  carbon  dioxide  (CO2)  is 
placed  in  the  lower  part  of  a  vessel  and  some  hydrogen  (H)  in  the 
upper  part,  it  will  be  found  after  leaving  them  for  a  moment  that 
they  are  mixed  due  to  diffusion;  i.e.,  there  will  be  a  large  percent- 
age of  carbon  dioxide  in  the  upper  portion  of  the  vessel,  notwith- 
standing the  fact  that  it  is  more  than  twenty  times  as  dense  as 
hydrogen.  Escaping  coal  gas  rapidly  diffuses  so  that  it  may  soon 
be  detected  in  any  part  of  the  room.  An  example  of  gas  Osmosis 
has  already  been  given  (see  Sec.  112). 

Since  gases  have  weight,  they  produce  pressure  for  the  same 
reason  that  liquids  do  (Sec.  113).  Thus  the  air  produces  what  is 
known  as  atmospheric  pressure,  which  is  about  15  Ibs.  per  sq.  in. 
In  the  case  of  illuminating  gas,  we  have  an  example  of  Transmis- 
sion of  Pressure  by  gas  from  the  gas  plant  to  the  gas  jet.  Another 
example  is  the  transmission  of  pressure  from  the  bicycle  pump  to 
the  bicycle  tire. 

Elasticity. — Gases,  like  liquids,  are  perfectly  elastic,  i.e., 
after  being  compressed  they  expand  to  exactly  their  original 
volume  upon  removal  of  the  added  pressure.  Gases  are  very 
easily  compressed  as  compared  with  liquids.  Indeed,  if  the  pres- 
sure upon  a  given  quantity  of  gas  is  doubled  or  trebled,  its  volume 
is  thereby  reduced  very  closely  to  1/2  or  1/3  its  original  volume, 
as  the  case  may  be.  The  fact  that  doubling  the  pressure  on  a 
certain  quantity  of  gas  halves  the  volume,  or,  in  general,  increas- 
ing the  pressure  n-fold  reduces  the  volume  to  \/n  the  original 
volume  provided  the  temperature  is  constant,  is  known  as  Boyle' 's 


PROPERTIES  OF  GASES  AT  REST  179 

Law.  This  very  important  gas  law  will  be  further  considered  in 
Sec.  139.  It  may  be  stated  that  Boyle's  law  does  not  apply 
rigidly  to  any  gas,  but  it  does  apply  closely  to  many  gases,  and 
through  wide  ranges  of  pressure. 

Expansibility. — Gases  possess  a  peculiar  property  not  possessed 
by  solids  or  liquids,  namely,  that  of  indefinite  expansibility 
(Sec.  98).  A  given  mass  of  any  gas  may  have  any  volume, 
depending  upon  the  pressure  (and  also  the  temperature)  to  which 
it  is  subjected.  If  the  pressure  is  reduced  to  1/10  its  original 
value  the  volume  expands  10-fold,  and  so  on.  A  mass  of  gas, 
however  small,  always  (and  instantly)  expands  until  it  entirely 
fills  the  enclosing  vessel. 

The  expansibility  and  also  the  compressibility  of  a  gas  may  be 
readily  shown  by  the  use  of  the  apparatus  sketched  in  Fig.  85. 
A  is  a  circular  brass  plate  which  is  perfectly 
flat  and  smooth  on  its  upper  surface.  B  is 
a  glass  bell  jar  turned  open  end  down 
against  A .  The  lower  edge  D  of  B  is  care- 
fully ground  to  fit  accurately  against  the 
upper  surface  of  A,  over  which  some  vase- 
line is  spread.  A  and  B  so  arranged  con-  ^ 
stitute  what  is  called  a  receiver.  The  re- 
ceiver forms  an  air-tight  enclosure  in  which 
is  placed  a  bottle  C,  across  the  mouth  of 
which  is  secured  a  thin  sheet  of  rubber  a,  thus  enclosing  some  air 
at  ordinary  atmospheric  pressure. 

By  means  of  the  pipe  E  leading  to  an  air  pump,  it  is  possible 
to  withdraw  the  air  from  the  space  H  within  the  receiver,  or  to 
force  air  into  the  space  H.  In  the  former  case  the  air  pressure 
in  H  is  reduced  so  as  to  be  less  than  one  atmosphere,  and  the  thin 
membrane  of  rubber  stretches  out  into  a  balloon-like  form  a\ ; 
while  in  the  latter  case,  that  is,  when  the  air  in  H  is  com- 
pressed, this  increased  pressure,  being  greater  than  the  pressure 
of  the  air  confined  in  C,  causes  the  membrane  to  assume  the  form 
a2.  The  process  by  which  the  air  pump  is  able  to  withdraw  from 
H  a  portion  of  the  air,  also  depends  upon  the  property  of  expansi- 
bility. A  reduction  of  pressure  is  produced  in  the  pump,  where- 
upon the  air  in  H  expands  and  rushes  out  at  E.  (This  process 
will  be  further  considered  in  Sees.  145  and  147.) 

Gas  Pressure  and  the  Kinetic  Theory. — According  to  the 
Kinetic  Theory  of  Gases  (Sec.  171),  the  pressure  which  a  «gas 


180  MECHANICS  AND  HEAT 

exerts  against  the  walls  of  the  enclosing  vessel  is  due  to  the  bom- 
bardment of  these  walls  by  the  gas  molecules  in  their  to-and-fro 
motion.  The  fact  that  the  ratio  of  the  densities  of  any  two  gases, 
e.g.,  carbon  dioxide  and  hydrogen,  when  subjected  to  the  same 
pressure  and  temperature,  is  the  same  as  the  ratio  of  their  mo- 
lecular weights,  shows  that  a  certain  volume  of  hydrogen  contains 
the  same  number  of  molecules  as  does  the  same  volume  of  carbon 
dioxide  or  any  other  gas  under  like  conditions  as  to  pressure  and  tem- 
perature. This  is  known  as  Avogadro's  Law.  It  will  be  recalled 
that  momentum  change  is  equal  to  the  impulse  required  to  pro- 
duce it  (Eq.  19,  Sec.  45).  Consequently,  since  the  hydrogen 
molecule  is  1/16  as  heavy  as  the  oxygen  molecule,  it  will  need  to 
have  4  times  as  great  velocity  as  the  oxygen  molecule  to  produce 
an  equal  contribution  toward  the  pressure.  For  each  impulse 
of  the  hydrogen  molecule  would  then  be  1/4  as  great  as  those  of 
the  oxygen  molecule,  but,  because  of  the  greater  velocity  of  the 
former,  these  impulses  would  occur  4  times  as  often. 

Knowing  the  density  of  the  gas,  it  is  comparatively  easy  to 
compute  the  molecular  velocity  required  to  produce  the  observed 
pressure.  The  average  velocity  of  the  hydrogen  molecule  at  0°  C. 
is,  on  the  basis  of  this  theory,  slightly  more  than  1  mi.  per  sec., 
while  that  of  the  oxygen  molecule  is  1/4  as  great,  as  already 
explained. 

The  very  rapid  diffusion  of  hydrogen  as  compared  with  other 
gases  would  be  a  natural  consequence  of  its  greater  velocity,  and 
therefore  substantiates  the  kinetic  theory.  The  observed 
increase  in  pressure  resulting  from  heating  confined  gases  is 
attributed  to  an  increase  in  the  average  velocity  of  its  molecules 
with  temperature  rise.  The  kinetic  theory  of  gas  pressure 
affords  a  very  simple  explanation  of  Boyles'  law  (close  of  Sec. 
139). 

132.  The  Earth's  Atmosphere. — Because  of  the  importance  and 
abundance  of  the  mixture  of  gases  known  as  air,  the  remainder  of 
the  chapter  will  be  devoted  largely  to  the  study  of  it.  It  may 
be  remarked  that  most  of  the  gases  are  very  much  like  air  with 
respect  to  the  properties  here  discussed. 

The  term  "atmosphere"  is  applied  to  the  body  of  air  that  sur- 
rounds the  earth.  Dry  air  consists  mainly  of  the  gases  nitrogen  and 
oxygen — about  76  per  cent,  of  the  former  and  23  per  cent,  of  the 
latter,  by  weight.  The  remaining  1  per  cent,  is  principally  argon. 
In  addition  to  these  gases  there  are  traces  of  other  gases,  impor- 


PROPERTIES  OF  GASES  AT  REST  181 

tant  among  which  are  carbon  dioxide  (C02)  and  water  vapor. 
The  amount  of  carbon  dioxide  in  the  air  may  vary  from  1  part  in 
3000  outdoors  (not  in  large  cities),  to  10  or  15  times  this  amount  in 
crowded  rooms.  The  oxygen  of  the  air  in  the  lungs  (see  Osmosis, 
Sec.  112)  is  partially  exchanged  for  carbon  dioxide  and  other 
impurities  of  the  blood;  as  a  result  the  exhaled  air  contains  4  or 
5  per  cent,  carbon  dioxide.  If  the  breath  is  held  for  an  instant 
and  then  carefully  and  slowly  exhaled  below  the  burner  of  a  lamp 
(the  hands  being  held  in  such  a  position  as  to  exclude  other  air 
from  the  burner),  the  flame  is  quickly  extinguished.  The  air  in 
this  case  does  not  have  enough  oxygen  to  support  combustion. 
Through  repeated  inhalation,  the  air  in  crowded,  poorly  venti- 
lated rooms  becomes  vitiated  by  carbon  dioxide.  Carbon  dioxide 
escapes  from  fissures  in  the  earth  and  forms  the  deadly  "choke 
damp"  of  mines.  It  also  results  from  the  explosion  of  "fire 
damp,"  or  marsh  gas  (CH4),  as  it  is  known  to  the  chemist.  If  a 
candle  when  carefully  lowered  into  a  shaft  is  extinguished  upon 
reaching  the  bottom,  the  presence  of  choke  damp  is  indicated. 

In  nature,  even  in  deserts,  air  never  occurs  dry.  The  amount 
of  water  vapor  in  the  air  varies  greatly,  sometimes  running  as 
high  as  1/2  oz.  per  cubic  yard  (about  1.5  per  cent.)  in  hot,  sultry 
weather.  As  moist  air  is  chilled,  its  ability  to  retain  water  vapor 
decreases  rapidly  and  precipitation  (Sec.  221)  occurs.  Conse- 
quently during  extremely  cold  weather  the  air  is  very  dry. 

133.  Height  of  the  Atmosphere. — As  meteors  falling  toward 
the  earth  strike  the  earth's  atmosphere,  the  heat  developed  by 
them  through  air  friction  as  they  rush  through  the  upper  strata 
of  rarefied  air  causes  them  to  become  quite  hot,  so  that  they 
shine  for  an  instant.  Suppose  that  one  is  seen  at  the  same  in- 
stant by  two  observers  40  or  50  miles  apart.  The  meteor  will 
appear  to  be  in  a  different  direction  from  one  observer  than  from 
the  other.  This  makes  possible  the  calculation  of  the  height  of 
the  point  at  which  the  meteor  began  to  glow.  But  it  could  not 
glow  before  striking  the  earth's  atmosphere;  hence  the  earth's 
atmosphere  extends  to  at  least  that  height. 

The  duration  of  twilight  after  sunset  also  enables  the  calcula- 
tion of  the  height  of  the  atmosphere.  Fine  dust  particles  float- 
ing in  the  upper  regions  of  the  air  are,  of  course,  flooded  with 
sunlight  for  a  considerable  time  after  sunset.  The  general  glow 
from  these  particles  constitutes  twilight.  If  an  observer  at  A 
(Fig.  85a)  looking  in  the  direction  AX  observes  the  last  trace  of 


182  MECHANICS  AND  HEAT 

twilight  when  it  is  sunset  at  B,  then  the  intersection  X  of  the 
tangents  at  A  and  B  is  the  highest  point  at  which  there  are  enough 
dust  particles  to  give  appreciable  twilight  effect. 

Knowing  the  angle  0  and  the  radius  of  the  earth,  the  height  of 
X  above  the  earth  is  readily  found.  Since  twilight  lasts  until 
the  sun  is  15  or  20  degrees  below  the  horizon,  we  see  that  9  is  15 
or  20  degrees.  If  6  is  18°,  X  is  about  50  miles  above  the  earth. 
Extremely  rare  air,  almost  free  from  dust  particles,  doubtless 
extends  far  above  this  height.  Estimates  of  the  height  of  the 
atmosphere  range  from  50  to  200  miles. 

The  upper  strata  of  air  are  very  rare  and  the  lower  strata 
comparatively  dense  due  to  compression  caused  by  the  weight 
of  the  air  above;  so  that  upon  a  mountain  3.5  miles  high  about 
half  of  the  weight  of  the  atmosphere  is  above  and  half  below. 
The  entire  region  above  7  miles  contains  only  1/4  of  the  earth's 
atmosphere. 


FIG.  85a. 

134.  Buoyant  Effect,  Archimedes'  Principle,  Lifting  Capacity  of 
Balloons. — Since  air  has  weight,  it  produces  a  certain  buoyant  effect 
just  as  liquids  do,  but  since  it  is  about  1/800  as  dense  as  water, 
the  buoyant  effect  is  only  1/800  as  great.  That  air  has  weight 
may  easily  be  shown  by  weighing  a  vessel,  e.g.,  a  brass  globe, 
first  with  air  in  it,  and  then  weighing  it  again  after  the  air  has 
been  partially  pumped  out  of  it  by  means  of  an  air  pump.  The 
difference  in  weight  is  the  weight  of  the  air  withdrawn.  Galileo 
(1564-1642)  weighed  a  glass  globe  when  filled  with  air  at  atmos- 
pheric pressure,  and  again  after  forcing  air  into  it.  The  observed 
increase  in  weight  he  rightly  attributed  to  the  additional  air 
forced  in. 

Archimedes'  Principle  (Sec.  118)  applies  to  gases  as  well  as  to 
liquids;  therefore  any  body  weighed  in  air  loses  weight  equal  to 
the  weight  of  the  air  displaced  by  the  body.  Thus  a  cubic  yard 


PROPERTIES  OF  GASES  AT  REST  183 

of  stone,  or  any  other  material,  weighs  about  two  pounds  less  in 
air  than  it  would  in  a  Vacuum,  i.e.,  in  a  space  from  which  all  air 
has  been  removed.  The  buoyant  force  exerted  by  the  air  upon  a 
150-lb.  man  is  about  3/16  lb.,  i.e.,  1/800X150  lb.;  since  his  body 
has  about  the  same  density  as  water.  Observe  that  he  would 
lose  practically  his  entire  weight  if  immersed  in  water;  hence,  since 
air  is  about  1/800  as  dense  as  water,  he  loses  1/800  of  his  weight 
by  being  immersed  in  air. 

The  lifting  capacity  of  a  balloon,  if  it  were  not  for  the  weight 
of  the  balloon  itself  and  the  contained  gas,  would  be  the  weight  of 
the  air  displaced,  or  approximately  2  Ibs.  for  each  cubic  yard  of 
the  balloon's  volume.  If  a  balloon  is  filled  with  a  light  gas,  e.g., 
with  hydrogen,  its  lifting  capacity  is  much  more  than  if  filled 
with  a  heavier  gas.  The  car  or  basket  attached  to  a  balloon 
contains  ballast,  which  may  be  thrown  overboard  when  the 
aeronaut  wishes  to  rise  higher.  When  he  wishes  to  descend  he 
permits  some  of  the  gas  to  escape  from  the  balloon,  thereby 
decreasing  the  volume  and  hence  the  weight  of  the  air  displaced. 

135.  Pressure  of  the  Atmosphere. — Since  the  air  has  weight, 
the  atmosphere  must  inevitably  exert  pressure  upon  all  bodies 
with  which  it  comes  in  contact.  This  pressure  at  sea  level  is 
closely  14.7  Ibs.  per  sq.  in.,  and  at  an  altitude  of  3.5  miles, 
about  half  of  this  value.  Ordinarily  the  atmospheric  pressure 
is  not  observable.  It  seems  hard  to  believe  that  the  human  body 
withstands  a  pressure  of  about  15  Ibs.  on  every  square  inch  of 
surface,  which  amounts  to  several  tons  of  force  upon  the  entire 
body,  without  its  even  being  perceptible.  It  is  certain,  however, 
that  such  is  the  case.  We  may  note  in  this  connection  that  the 
cell  walls  in  the  tissues  of  the  body  do  not  have  to  sustain  this 
pressure,  since  the  cells  are  filled  with  material  at  this  same  pres- 
sure. Thus,  the  atmospheric  pressure  of  about  15  Ibs.  per  sq.  in. 
has  no  tendency  to  crush  the  lung  cells  when  they  are  filled  with 
air  at  this  same  pressure.  Sudden  changes  in  pressure,  however, 
such  as  accompany  rapid  ascent  or  descent  in  a  balloon,  or  in  a 
diving  bell,  produce  great  discomfort. 

The  pressure  exerted  by  water  at  a  depth  of  about  34  ft.  is  one 
atmosphere  (Sec.  136),  so  that  a  diver  34  ft.  below  the  surface  of  a 
lake  experiences  a  pressure  of  2  "atmospheres,"  one  atmosphere 
due  to  the  air,  and  one  due  to  the  water.  Divers  can  work  more 
than  100  ft.  beneath  the  surface  of  water,  and  must  then  experi- 
ence a  pressure  of  4  or  5  atmospheres,  i.e.,  60  or  75  Ibs.  per  sq. 


184 


MECHANICS  AND  HEAT 


in.     The  air  which  the  diver  breathes  must,  under  these  circum- 
stances, be  also  under  this  same  high  pressure. 

The  pressure  of  the  atmosphere  acts  in  a  direction  which  is  at 
all  points  perpendicular  to  the  surface  of  a  body  immersed  in  it. 
Compare  the  similar  behavior  of  liquids  (Sec.  114).  That  the 
atmospheric  pressure  may  be  exerted  vertically  upward,  and  that 
it  may  be  made  to  lift  a  heavy  weight,  is  forcibly  shown  by  the 
following  experiment. 

A  cylinder  A,  having  a  tight-fitting  piston  P  to  which  is 
attached  the  weight  W,  is  supported  as  shown  (Fig.  86).  If, 
by  means  of  an  air  pump  connected  to  the  tube  C,  the  air  is 
partly  withdrawn  from  the  space  B,  it  will  be  found 
that  P  will  rise  even  if  W  is  very  heavy.  If  it  were 
possible  to  remove  all  of  the  air  from  B,  producing 
in  the  cylinder  a  perfect  vacuum,  the  pressure  within 
the  cylinder,  and  hence  the  downward  pressure  on  P 
would  be  zero.  The  upward  pressure  upon  P,  be- 
ing atmospheric  pressure  or  about  14.7  Ibs.  per  sq. 
^  c  \.  in.,  would  enable  it  to  lift  147  Ibs.,  provided  it  had 

an  area  of  10  sq.  in. 

If  only  part  of  the  air  is  withdrawn  from  B,  so 
that  the  pressure  within  the  cylinder  is  say  5  Ibs. 
per  sq.  in.,  P  would  then  exert  a  lifting  force  of  14.7 
minus  5,  or  only  9.7  Ibs.  for  each  square  inch  of  its 
surface.  The  pressure  of  the  atmosphere  cannot 
be  computed  by  use  of  the  formula  p  =  hdg;  because 
the  height  is  uncertain,  and  also  because  the  density  d  varies,  be- 
ing much  less  at  high  altitudes.  The  pressure  is  very  easily 
obtained,  however,  by  means  of  the  barometer  described  in  the 
next  section. 

136.  The  Mercury  Barometer. — There  are  several  different 
kinds  of  barometers.  The  simplest,  and  also  the  most  accurate 
form  is  shown  in  Fig.  87.  Various  devices  found  in  the  practical 
instrument  for  making  adjustments,  and  for  determining  very 
accurately  the  height  of  the  mercury  column  (vernier  attach- 
ment), are  omitted  in  the  sketch  for  the  sake  of  simplicity  in 
showing  the  essentials  and  in  explaining  the  principle  involved. 
A  glass  tube  A,  about  1/3  in.  in  diameter  and  3  ft.  in  length, 
and  closed  at  the  end  a,  is  filled  with  mercury,  and  then,  a 
stopper  being  held  against  the  open  end  to  prevent  any  mercury 
from  escaping,  it  is  inverted  and  placed  open  end  down  in  a  vessel 


FIG.  86. 


PROPERTIES  OF  GASES  AT  REST 


185 


of  mercury  B,  as  shown.  Upon  removing  the  stopper,  it  might 
be  expected  that  the  mercury  would  run  out  until  it  stood  at  the 
same  height  inside  and  outside  the  tube.  Indeed  it  would  do  this 
if  there  were  at  a  the  slightest  aperture  to  admit  the  air  to  the 
upper  portion  of  the  tube,  for  then  the  pressure  inside  and  out- 
side the  tube  would  be  exactly  the  same,  namely,  atmospheric 
pressure.  If  a  is  perfectly  air-tight,  it  will  be  found  that  some 
mercury  runs  out  of  the  tube  until  the  upper  surface  sinks  to  a 
point  c.  The  height  h  of  the  mercury  column  c  to  6,  is  called 
the  Barometric  Height,  and  is  usually  about  30  in.  near  sea  level. 
Evidently  the  space  a  to  c  contains  no  air  nor 
anything  else.  Such  a  space  is  called  a  Vacuum. 
The  downward  pressure  on  the  surface  of  the  mer- 
cury at  c  is  then  zero. 

This  experiment  was  first  performed  in  1643 
by  Torricelli  (1608-1647)  and  is  known  as  Tarri- 
celli's  experiment.  A  few  years  later  the  French 
physicist  Pascal  (1623-1662)  had  the  experiment 
performed  on  a  mountain,  and  found,  as  he  had 
anticipated,  that  the  column  be  was  shorter  there 
than  at  lower  altitudes. 

Consider  the  horizontal  layer  of  mercury  particles 
6  within  the  tube  and  on  the  same  level  as  the  sur- 
face s  outside  the  tube.     The  downward  pressure 
on  this  layer  is  hdg  in  which  h  is  the  height  of  the 
column  be,  and  d  is  the  density  of  mercury  (13.596 
gm.  per  cm.3).     But  the  upward  pressure  on  this 
layer  6  must  have  this  same  value,  since  the  layer  is 
in  equilibrium.     The  only  cause  for  this  upward 
pressure,  however,  is  the  pressure  of  the  atmosphere  upon  the 
surfaces  of  the  mercury,  which  pressure  is  transmitted  by  the 
mercury  to  the  inside  of  the  tube.     Hence  the  pressure  of  the  at- 
mosphere is  equal  to  the  pressure  exerted  by  the  mercury  column, 
or  hdg.     The  barometric  height  varies  greatly  with  change  of 
altitude  and  also  considerably  with  change  of  weather.     Stand- 
ard atmospheric  pressure  supports  a  column  of  mercury  76  cm. 
in  height,  at  latitude  45°  and  at  sea  level  (0  =  980.6);  hence 
Standard  Atmos.  Pr.  =hdg  =  76X13.596X980.6=  1,000,000  dynes 
per  sq.  cm.  (approx.). 
This  is  approximately  14.7  Ibs.  per  sq.  in. 

Quite  commonly  the  pressure  of  the  atmosphere  is  expressed 


FIG.  87. 


186  MECHANICS  AND  HEAT 

simply  in  terms  of  the  height  of  the  barometric  column  which  it 
will  support,  as  "29.8  in.  of  mercury,"  "74  cm.  of  mercury."  At 
sea  level  the  pressure  of  the  atmosphere  is  usually  about  30  in. 
of  mercury;  at  an  altitude  of  3.5  mi.,  about  15  in.  of  mercury; 
while  aeronauts  at  still  higher  altitudes  have  observed  as  low  a 
barometric  height  as  9  in. 

Unless  great  care  is  taken  in  filling  the  tube  (Fig.  87),  it  will 
be  found  that  some  air  will  be  mixed  with  the  mercury,  and  that 
therefore  the  space  from  a  to  c,  instead  of  containing  a  vacuum, 
will  contain  some  air  at  a  slight  pressure.  This  counter  pressure 
will  cause  the  mercury  column  to  be  somewhat  shorter  than  it 
otherwise  would  be,  and  the  barometer  will  accordingly  indicate 
too  low  a  pressure.  If  the  mercury  is  boiled  in  the  tube  before 
inverting,  the  air  will  be  largely  driven  out  and  the  error  from  this 
source  will  be  greatly  reduced.  It  will  be  evident  that  this  slight 
counter  pressure  of  the  entrapped  air,  in  case  a  trace  of  air  is  left 
in  the  space  ac,  plus  hdg  for  the  column  of  mercury  be,  gives  the 
total  downward  pressure  at  b.  But  this  total  pressure  must  equal 
the  upward  pressure  at  6,  due  to  the  atmosphere  as  shown. 
Hence  hdg  will  give  a  value  for  the  atmospheric  pressure,  which  is 
too  small  by  exactly  the  amount  of  pressure  on  c,  due  to  the 
entrapped  air. 

Since  water  is  only  1/13.6  times  as  dense  as  mercury,  it  follows 
that  atmospheric  pressure  will  support  13.6  times  as  long  a  col- 
umn of  water  as  of  mercury,  or  about  13.6X30  in.,  which  is 
approximately  34  ft.  Accordingly,  the  pressure  required  to 
force  water  through  pipes  a  vertical  height  of  340  ft.  is  approxi- 
mately 10  atmospheres,  or  150  Ibs.  per  sq.  in.,  in  addition  to  the 
pressure  required  to  overcome  friction  in  the  pipes. 

137.  The  Aneroid  Barometer. — The  Aneroid  Barometer  con- 
sists of  an  air-tight  metal  box  of  circular  form  having  a  corrugated 
top  and  containing  rarefied  air.  As  the  pressure  of  the  atmos- 
phere increases,  the  center  of  this  top  is  forced  inward,  and  when 
the  pressure  decreases  the  center  moves  outward,  due  to  the 
elasticity  of  the  metal.  This  motion  of  the  center  is  very  slight 
but  is  magnified  by  a  system  of  levers  connecting  it  with  a  pointer 
that  moves  over  the  dial  of  the  instrument.  The  position  of  this 
pointer  upon  the  dial  at  a  time  when  the  mercury  column  of  a 
simple  barometer  is  75  cm.  high  is  marked  75,  and  so  on  for 
other  points.  This  type  of  barometer  is  light,  portable,  and 
easily  read. 


PROPERTIES  OF  GASES  AT  REST  187 

138.  Uses  of  the  Barometer. — Near  a  storm  center  the  atmos- 
pheric pressure  is  low  (Sec.  225),  consequently  a  falling  barom- 
eter indicates  an  approaching  storm.     Knowing  the  barometric 
readings  at  a  great  number  of  stations,  the  Weather  Bureau  can 
locate  the  storm  centers  and  predict  their  probable  positions  a 
few  days  in  advance.     Thus  this  Bureau  is  able  to  furnish  infor- 
mation which  is  especially  valuable  to  those  engaged  in  shipping. 

Due  to  the  capricious  character  of  the  weather,  these  predic- 
tions are  not  always  fulfilled.  Although  the  forcasting  of  the 
weather  a  year  in  advance  is  absolute  nonsense,  there  are  many 
who  have  more  or  less  faith  in  such  forcasts.  Of  course  one  is 
fairly  safe  in  predicting  "cold  rains"  for  March,  "hot  and  dry" 
for  August,  etc.,  but  to  fix  a  month  or  a  year  in  advance  the  date 
of  a  storm  from  the  study  of  the  stars  (which  certainly  have 
nothing  to  do  with  the  weather),  is  surely  out  of  place  in  this 
century. 

As  stated  in  Sec.  136,  the  barometric  height  decreases  as  the 
altitude  increases.  Near  sea  level  the  rate  of  this  decrease  is 
about  0.1  in.  for  each  90  ft.  of  ascent.  At  higher  altitudes  this 
decrease  is  not  so  rapid  because  of  the  lesser  density  of  air  in  those 
regions.  A  formula  has  been  developed,  by  the  use  of  which  the 
mountain  climber  can  determine  his  altitude  fairly  well  from  the 
readings  of  his  barometer.  An  "altitude  scale"  is  engraved  on 
many  aneroid  barometers,  by  means  of  which  the  altitude  may  be 
roughly  approximated. 

139.  Boyle's  Law. — The  volume  of  a  given  mass  of  gas,  mul- 
tiplied by  the  pressure  to  which  it  is  subjected,  is  found  to  be 
nearly  constant  if  the  temperature  remains  unchanged.     This 
is  known  as  Boyle's  Law  and  may  be  written 

pV  (temp,  constant)  =K  (72) 

This  important  law  was  discovered  by  Robert  Boyle  (1627- 
1691)  and  published  in  England  in  1662.  Fourteen  years  later 
it  was  rediscovered  by  the  French  physicist  Marriotte.  This 
illustrates  the  slow  spread  of  scientific  knowledge  in  those  days. 
In  France  it  is  called  Marriotte's  Law. 

From  the  equation  it  may  be  seen  that  to  cause  a  certain  vol- 
ume of  gas  to  shrink  to  1/n  its  original  volume  will  require  the 
pressure  to  be  increased  n-fold,  provided  that  the  temperature 
remains  constant.  The  equation  also  shows  that  if  we  permit 
a  certain  mass  of  confined  gas  to  expand  to,  say,  10  times  its 


188  MECHANICS  AND  HEAT 

original  volume,  then  the  new  pressure  will  be  1/10  as  great  as 
the  original  pressure.  By  original  pressure  and  volume  we  mean 
the  pressure  and  volume  before  expansion  occurred.  As  already 
stated,  Boyle's  Law  applies  closely  to  many  gases,  rigidly  to 
none. 

To  illustrate  Boyle's  Law  by  a  problem,  let  P  (Fig.  88)  be 
an  air-tight,  frictionless  piston  of,  say,  10  sq.  in.  surface  and  of 
negligible  weight,  enclosing  in  vessel  A  a  quantity  of  air  at  atmos- 
pheric pressure,  say  15  Ibs.  per  sq.  in.'  Let  it  be  required  to  find 
how  heavy  a  weight  must  be  placed  upon  P  to  force  it  down  to 
position  PI,  thereby  compressing  the  entrapped  air  to  1/3  its 
original  volume. 

From  Eq.  72,  we  see  that  the  pressure  of  the  entrapped  air  in 
the  latter  case  will  be  increased  3-fold  and  hence  will  exert  upon  P 
when  at  PI,  an  upward  pressure  of  45  Ibs.  per  sq.  in. 
The  outside  atmosphere  exerts  a  pressure  of  15  Ibs. 
per  sq.  in.  on  P;  consequently  the  remaining  30 
Ibs.  pressure  required  to  hold  P  down  must  be 
furnished  by  the  added  weight.  A  pressure  of  30 
Ibs.  per  sq.  in.  over  a  piston  of  10  sq.  in.  surface 
amounts  to  300  Ibs.  force;  hence  the  added  weight 
required  is  300  Ibs. 


I    t 


FIG.  88.  We  may  explain  Boyle's  Law  in  full  accord  with 

the  Kinetic  Theory  of  gas  pressure  (Sec.  131).  For 
when  the  volume  of  the  air  in  the  vessel  represented  in  Fig.  88  is 
reduced  to  1/3  its  original  volume,  the  molecules,  if  they  con- 
tinue to  travel  at  the  same  velocity,  would  strike  the  piston 
three  times  as  frequently,  and  experience  each  time  the  same 
amount  of  momentum  change,  as  in  the  original  condition. 
They  would  therefore  produce  three  times  as  great  pressure 
against  the  piston  as  they  did  in  the  original  condition,  which, 
it  will  be  noted,  accords  with  experimental  results. 

140.  Boyle's  Law  Tube,  Isothermals  of  a  Gas. — A  bent  glass 
tube  A  (Fig.  89),  having  the  short  arm  closed  at  e,  and  the  long 
arm  open  and  terminating  in  a  small  funnel  at  6,  is  very  conven- 
ient to  use  in  the  verification  of  Boyle's  Law.  The  method  of 
performing  the  experiment  is  given  below. 

A  few  drops  of  mercury  are  introduced  into  the  tube  and  ad- 
justed until  the  mercury  level  c  in  the  long  arm  is  at  the  same 
height  as  the  mercury  level  d  in  the  short  arm.  As  more  mer- 
cury is  poured  into  the  tube  at  6,  the  pressure  on  the  air  enclosed 


PROPERTIES  OF  GASES  AT  REST 


189 


in  de  is  increased,  which  causes  a  proportional  decrease  in  its 
volume. 

If  now  we  plot  these  values  of  the  pressure  as  ordinates  (Sec. 
41)  and  the  corresponding  values  of  the  volume  as  abscissae, 
we  obtain,  provided  the  room  temperature  is  20°,  the  curve 
marked  20°  in  Fig.  90.  This  is  called  the  Isothermal  for  air  at 
20°  C. 

Method  in  Detail. — If  the  barometer  reads  75  cm.,  that  is,  if 
the  atmospheric  pressure  is  75  cm.  of  mercury,  then,  since  c  and 
d  are  at  the  same  level,  it  follows  that  the  pressure  on  the  en- 
trapped air  is  75  cm.  of  mercury.  If  the  tube  has  1  sq.  cm.  cross 


\dt 


FIG.  89. 

section  and  de  is  20  cm.,  then  the  corresponding  volume  of  the 
air  is  20  cm.3  Accordingly  the  point  marked  A  on  the  curve 
(ordinate  75,  abscissa  20)  represents  the  initial  state  of  the 
entrapped  air.  Next,  mercury  is  poured  into  6  until  it  stands  at 
Ci  and  di  in  the  tubes.  If  the  vertical  distance  from  c\  to  d\  is 
25  cm.,  the  pressure  upon  the  air  in  d\e\  will  be  25  cm.  more  than 
atmospheric  pressure,  or  a  total  of  100  cm.  Since  this  is  4/3 
of  the  initial  pressure,  the  corresponding  volume  should  be  3/4 
of  the  initial  volume,  or  15  cm.3  Measurement  will  show  that 
dtfi  is  15  cm.3  Hence  point  B  (ordinate  100  and  abscissa  15) 


190 


MECHANICS  AND  HEAT 


represents  the  new  state  of  the  entrapped  air  as  regards  its  pres- 
sure and  volume.  When  still  more  mercury  is  poured  in,  the 
mercury  stands  at,  say,  c2  and  d%,  the  vertical  distance  c2c?2  being 
75  cm.  The  pressure  upon  the  entrapped  air  (dzez)  is  now  this 
75  cm.  plus  atmospheric  pressure,  or  a  total  of  150  cm.  Since 
this  pressure  is  twice  the  initial  pressure,  the  corresponding  vol- 
ume is,  as  we  should  expect,  one-half  the  original  volume,  or 
10  cm.3  Hence  the  point  on  the  curve  marked  C  (ordinate  150, 
abscissa  10)  represents  this,  the  third  state  of  the  entrapped  air. 
To  obtain  smaller  pressures  than  one  atmosphere,  a  different 
form  of  apparatus  shown  at  the  right  in  Fig.  89  is  more  conven- 


60  80  100 

VOLUME  IN  CM.' 
FIG.  90. 


ient.  A  small  tube  B  of,  say  1  sq.  cm.  cross  section,  is  filled  with 
mercury  to  within  20  cm.  of  the  top  and  then  stoppered  and 
inverted  in  a  large  tube  C  which  is  nearly  filled. with  mercury. 
Upon  removing  the  stopper  and  pressing  the  tube  down  until  the 
mercury  in  both  tubes  stands  at  the  same  height,  it  will  be  seen 
that  the  volume  of  the  entrapped  air  (which  is  now  at  atmospheric 
pressure)  is  20  cm.3  If,  now,  tube  B  is  raised  until  the  mercury 
within  it  stands  at  d3,  and  if  d3f  is  25  cm.,  then  the  pressure  upon 
the  entrapped  air  is  50  cm. ;  for  this  pressure  plus  the  pressure  of 
the  column  of  mercury  d3f  must  balance  the  atmospheric  pressure 
of  75  cm.  Since  this  pressure  (50  cm.)  is  2/3  of  the  initial  pres- 


PROPERTIES  OF  GASES  AT  REST  191 

sure,  the  corresponding  volume  in  accordance  with  Boyle's  Law 
must  be  3/2  of  the  original  volume,  or  30  cm.3  Measurement 
will  show  that  d&z  is  30  cm.  Hence  the  point  on  the  curve 
marked  D  (ordinate  50,  abscissa  30)  represents  this  state  of  the 
air.  If  tube  B  is  raised  still  farther  until  the  mercury  within  it 
stands  50  cm.  higher  than  in  C,  then  the  pressure  of  the  entrapped 
air  is  25  cm.,  or  1/3  of  the  initial  pressure,  and  its  volume  will  be 
found  to  be  three  times  the  initial,  or  60  cm.3  Hence  point  E 
(ordinate  25,  abscissa  60)  represents  this,  the  fifth  state  of  the 
entrapped  air.  In  the  same  way  points  F,  G,  etc.,  are  determined. 
Drawing  a  smooth  curve  through  these  points  A,  B,  (7,  etc., 
gives  the  isothermal  for  air  at  20°  C.  When  we  take  up  the  study 
of  heat  we  will  readily  see  that  the  100°  isothermal  would  be 
drawn  about  as  shown  (see  dotted  curve). 

Observe  that  the  three  rectangles,  A-75-0-20,  D-50-0-30, 
and  #-25-0-60  all  have  the  same  area  and  that  this  area  repre- 
sents the  product  of  the  pressure — 75,  50,  or  25  as  the  case  may 
be,  and  the  corresponding  volumes  of  the  entrapped  air  for  the 
three  different  states  which  are  represented  respectively  by  the 
points  A,  D,  and  E  on  the  curve.  Thus  the  curve  verifies  Boyle's 
Law  as  expressed  in  Eq.  72,  and  shows  that  the  constant  K  in 
this  equation  is,  for  this  particular  amount  of  gas,  1500;  for  75  X  20, 
50X30,  and  25X60,  each  gives  1500. 

141.  The  Manometers  and  the  Bourdon  Gage. — Manome- 
ters are  of  two  kinds,  the  Open  Tube  Manometer,  usually  used 


FIG.  91. 

for  measuring  small  differences  in  pressure,  and  the  Closed 
Tube  Manometer  which  may  be  used  to  measure  the  total  pres- 
sure to  which  a  gas  or  a  liquid  is  subjected. 

The  Open  Tube  Manometer  (Fig.  91)  consists  of  a  U-shaped 
glass  tube  T,  open  at  both  ends  and  containing  some  liquid, 
frequently  mercury.  If,  when  the  manometer  is  connected  with 
the  vessel  A  containing  some  gas,  it  is  found  that  the  mercury 
stands  at  the  same  height  in  both  arms,  namely,  at  a  and  b,  then 


192 


MECHANICS  AND  HEAT 


FIG.  92. 


the  pressure  of  this  gas  which  acts  upon  a,  must  be  equal  to  the 
pressure  of  the  atmosphere  which  acts  upon  6.  If  the  mercury 
meniscus  61  is  higher  than  a\  by  a  distance  hi  cm.,  then  the  pres- 
sure in  B  is  1  atmosphere  +  hidg  dynes  per  cm.2,  in  which  d  is 
the  density  of  the  mercury.  The  pressure  of  the  gas  in  C  is 
evidently  less  than  one  atmosphere  by  the  amount  hzdg.  If  very 
small  differences  in  pressure  are  to  be  meas- 
ured it  is  best  to  employ  a  light  liquid  for 
the  manometer.  « 

The  Closed  Tube  Manometer  (Fig.  92)  may 
be  used  for  measuring  high  pressures,  such 
as  the  pressure  of  steam  in  steam  boilers, 
city  water  pressure,  etc.  Let  D  represent 
a  steam  boiler  containing  some  water,  and 

T,  an  attached  closed  tube  manometer.  If  the  mercury  stands 
at  the  same  height  in  both  arms  a  and  6  when  valves  leading 
from  D  to  the  outside  air  are  open,  it  shows  that  the  entrapped 
air  in  the  manometer  is  at  one  atmosphere  pressure.  If,  upon 
closing  these  valves  and  heating  the  water  in  D,  the  pressure  of 
the  steam  developed  forces  the  mercury  down  to  a'  in  the  left 
arm  and  up  to  6'  in  the  right  arm, 
thereby  reducing  the  volume  of  the 
entrapped  air  to  1/3  its  original 
volume,  it  follows  from  Boyle's  Law 
that  the  pressure  on  it  is  increased 
3-fold  and  is  therefore  3  atmospheres. 
The  steam  in  D  is  then  at  3  atmos- 
pheres pressure.  It  is  really  slightly 
more  than  this,  for  the  mercury 
stands  a  distance  h  higher  in  the 
right  arm  than  in  the  left.  The 
correction  is  clearly  hdg.  That  is, 
the  pressure  upon  the  enclosed  air 
above  b'  would  be,  under  these  cir- 
cumstances, exactly  3  atmospheres 

while  the  steam  pressure  in  the  boiler  would  be  3  atmospheres 
plus  the  pressure  hdg  due  to  the  mercury  column  of  height  h. 

The  Bourdon  Gage. — The  essentials  of  the  Bourdon  gage,  which 
is  widely  used  for  the  measurement  of  steam  pressure  and  water 
pressure,  are  shown  in  Fig.  93.  The  metal  tube  T,  which  rs 
closed  at  B,  is  of  oval  cross  section,  CD  being  the  smaller  diameter. 


FIG.  93. 


PROPERTIES  OF  GASES  AT  REST  193 

If  A  is  connected  to  a  steam  boiler,  the  pressure  of  the  steam 
causes  the  cross  section  of  the  tube  to  become  more  nearly  cir- 
cular, i.e.,  it  causes  the  smaller  diameter  CD  to  increase.  Ob- 
viously, pushing  the  sides  C  and  D  of  the  tube  farther  apart  will 
cause  the  tube  to  straighten  slightly,  thereby  moving  B  to  the 
right  and  causing  the  index  7  to  move  over  the  scale  as  indicated. 
By  properly  calibrating  the  gage,  it  will  read  directly  the  steam 
pressure  in  pounds  per  square  inch.  Most  steam  gages  are  of 
this  type.  The  same  device  may  be  used  to  measure  the  pressure 
of  water,  or  the  pressure  of  any  gas. 

The  Vacuum  Manometer  or  Vacuum  Gage. — If  the  space  above  b  in 
tube  T  (Fig.  92)  were  a  perfect  vacuum  (e.g.,  if  that  arm  of  the  tube  were 
first  entirely  filled  with  mercury),  and  if  nearly  all  of  the  air  were 
pumped  out  of  D,  then  T  would  be  a  "vacuum"  gage.  If,  under  these 
circumstances,  meniscus  6  stood  0.05  mm.  higher  than  a,  it  would  show 
that  the  pressure  of  the  remnant  of  the  air  in  D  was  only  equal  to  that 
produced  by  a  column  of  mercury  0.05  mm.  in  height.  If  the  "  vacuum  " 
in  D  were  perfect,  then  a  and  6  would  stand  at  the  same  height. 

PROBLEMS 

1.  What  is  the  pressure  of  the  atmosphere  (in  dynes  per  cm.2)  when  the 
mercury  barometer  reads  74.2  cm.? 

2.  What  is  the  pressure  of  the  atmosphere  (in  Ibs.  per  in.2)  when  the  ba- 
rometer reads  28.2  in.? 

3.  If,  in  Fig.  89,  d3/  =  30  cm.  and  the  barometer  reads  74  cm.,  what  is  the 
pressure  on  the  entrapped  air  in  centimeters  of  mercury?     In  atmospheres? 

4.  An  aneroid  barometer,  at  a  certain  time,  reads  29.9  in.  at  sea  level  and 
29.35  in.  on  a  nearby  hill.     What  is  the  approximate  altitude  of  the  hill? 
(Sec.  138.) 

6.  The  liquid  (oil  of  density  0.9  gm.  per  cm.3)  in  an  open  tube  manometer 
stands  4  cm.  higher  in  the  arm  which  is  exposed  to  the  confined  gas  than  it 
does  in  the  other  arm.  What  is  the  pressure  exerted  by  the  gas?  The 
barometric  reading  is  29  in. 

6.  A  closed  tube  manometer  contains  an  entrapped  air  column  8  cm.  in 
length  when  exposed  to  atmospheric  pressure,  and  3.2  cm.  in  length  when 
connected  to  an  air  pressure  system.     What  is  the  pressure  of  the  system? 
The  mercury  stood  at  the  same  level  in  both  arms  in  the  first  test. 

7.  If  a  1000-lb.  weight  is  rested  upon  P  (Fig.  88),  what  will  be  the  new 
volume  of  the  enclosed  air  in  terms  of  the  old? 

8.  A  certain  balloon  has  a  volume  equal  to  that  of  a  sphere  of  15-ft. 
radius.     What  weight,  including  its  own,  will  it  lift  when  the  density  of  the 
air  is  (a)  2  Ibs.  per  cubic  yard?     (b)  0.0011  gm.  per  cm.3?     Express  the 
weight  in  pounds  in  both  cases. 

9.  Plot  a  curve  similar  to  that  shown  in  Fig.  90  and  explain  how  it  is 
obtained. 


CHAPTER  XII 
PROPERTIES  OF  FLUIDS  IN  MOTION 

142.  General  Discussion. — The  steady  flow  of  a  fluid,  either  a 
liquid  or  a  gas,  at  a  uniform  velocity  through  a  level  pipe  from  one 
point  to  another,  is  always  due  to  a  difference  in  pressure  main- 
tained between  the  two  points  (friction  head,  see  footnote).  This 
difference  in  pressure  multiplied  by  the  cross  section  of  the  pipe 
gives  the  total  force  which  pushes  the  column  of  fluid  through  the 
pipe.  Since  the  velocity  of  this  column  is  neither  increasing  nor 
decreasing,  there  is  no  accelerating  force,  and  the  above  pushing 
force  must  be  just  equal  to  the  friction  force  exerted  upon  the 
column  by  the  pipe.  If  at  any  point  the  fluid  is  increasing  in 
velocity,  an  accelerating  force  F  must  be  present,  and  part  of  the 
pressure  difference  (velocity  head)1  is  used  in  producing  this 
accelerating  force.  F  is  equal  to  the  mass  M  of  the  liquid  being 
accelerated,  multiplied  by  its  acceleration  a  (Sec.  25,  F  =  Ma), 

Just  as  the  canal  boat  (Sec.  43),  by  virtue  of  its  inertia,  develops 
a  forward  driving  inertia  force  (F  =  Ma)  which  pushes  it  onto  the 

1  Head  of  Water. — In  hydraulics,  the  pressure  at  a  point,  or  the  difference 
in  pressure  between  two  points,  is  called  pressure  head,  and  is  measured  in 
terms  of  the  height  (in  feet)  of  the  column  of  water  required  to  produce 
such  pressure,  or  pressure  difference.  To  illustrate,  suppose  that  in  certain 
hydraulic  mining  operations,  the  supply  reservoir  is  600  ft.  above  the  hose 
nozzle,  and  that  the  velocity  of  the  water  as  it  leaves  the  nozzle  is  100  ft. 
per  sec.  Since  a  body  must  fall  about  150  ft.  to  acquire  a  velocity  of  100  ft. 
per  sec.,  the  head  required  to  impart  this  velocity  to  the  water  would  be 
150  ft.  (see  Sec.  143).  Consequently  the  Velocity  Head  required  is  150  ft. 
The  remainder  of  the  600-ft.  head,  namely  450  ft.,  is  used  in  overcoming 
friction  in  the  conveying  pipes  and  hose,  and  is  called  Friction  Head.  As 
the  water  from  the  reservoir  enters  the  conveying  pipes  it  must  acquire 
velocity.  As  the  water  passes  from  the  pipe  into  the  much  smaller  hose, 
and  again  as  it  passes  from  the  hose  into  the  tapering  nozzle,  it  must  ac- 
quire additional  velocity.  Thus  the  total  head  of  600  ft.  is  equal  to  the  sum 
of  the  velocity  heads  of  the  pipe,  the  hose,  and  the  nozzle,  in  addition  to 
the  friction  head  for  all  three.  If  the  size  of  the  conveying  pipe  or  hose 
changes  abruptly  (either  increases  or  decreases)  eddies  will  be  formed  which 
cause  considerable  friction  and  consequent  loss  of  head.  To  reduce  this 
loss,  the  pipe  should  flare  as  it  enters  tie  reservoir. 

194 


PROPERTIES  OF  FLUIDS  IN  MOTION 


195 


sand  bar;  so  also  a  moving  fluid  (e.g.,  water,  steam,  or  air)  exerts 
a  driving  inertia  force  (F  =  Ma)  against  any  body  that  changes 
its  velocity.  It  is  this  inertia  force  which  drives  the  wind  mill, 
the  steam  turbine,  and  the  turbine  water  wheel,  or  any  other  water 
wheel  which  utilizes  the  velocity  of  the  water. 

A  thorough  understanding  of  the  above  principles  and  their 
applications  gives  one  a  fair  elementary  knowledge  of  the  subject 
of  Hydraulics.  A  discussion  of  Fig.  94  will  aid  in  securing  such  an 
understanding. 

Let  B  be  a  level  water  pipe  communicating  with  the  vertical 
pipes  C,  E,  and  F,  and  with  the  tank  A.  If  B  is  closed  at  G,  so 
that  no  water  flows  through  it,  the  water  will  stand  at  the  same 
level,  say  at  a-d-e-f,  in  the  tank  and  in  the  vertical  pipes.  If 
G  is  removed  the  water  will  at  first  flow  out  slowly,  for  it  will 


D 

C                     j 

E                      j 

a 

'd 

c 

e 

Y 

HH 

jf 

*  c' 

Pipii 

= 

:i       .^S- 

e' 

—-=--=— 

- 

i 

| 

C 

t    **"•-  ^. 

L  ^-.^ 

—  _  

FIG.  94. 

take  the  force  at  i,  due  to  the  tank  pressure,  a  short  interval  of 
time  to  impart  to  all  of  the  water  in  pipe  B  a  high  velocity.  After 
a  few  seconds  the  water  will  be  flowing  rapidly  and  steadily  at  G, 
and  the  water  in  the  vertical  pipes  will  stand  at  the  different 
levels  c',  d,  e',  and  /'.  Observe  that  c',  e',  f,  and  G  all  lie 
in  the  same  straight  line.  This  uniform  pressure  drop  or  friction 
head  loss  is  due  to  the  fact  that  the  friction  is  the  same  in  all 
parts  of  B.  If  the  pipe  B  between  c"  and  e"  were  rusty '  and 
rough,  or  smaller  than  elsewhere,  the  friction  head  between  these 
two  points  would  be  greater  than  elsewhere,  causing  e'  to  be 
lower  than  shown.  In  such  case  c',  e' ',  and  f  would  not  lie 
in  a  straight  line. 

Observe  that  removing  G  has  not  lowered  the  level  in  pipe  D, 
but  has  produced  a  decided  drop  in  C.  This  difference  in  level 
(hz),  corresponding  to  a  difference  in  pressure  of  h^dg,  cannot  be 


196  MECHANICS  AND  HEAT 

due  to  friction  head  in  the  short  distance  ic".  This  difference  in 
pressure  is  mainly  due  to  the  pressure  head  (velocity  head) 
required  to  accelerate  the  water  as  it  passes  from  the  tank,  where 
it  is  almost  without  motion,  to  the  pipe  B,  where  it  moves  rapidly. 
If  the  pipe  B  were  nearly  closed  at  G,  the  flow  would  be  slow, 
and  the  friction  head  small,  so  that  the  water  would  stand  nearly 
as  high  in  C,  E,  and  F  as  in  the  tank  A.  Just  as  the  heavy  flow 
from  G  lowers  the  water  pressure  at  /"  and  hence  in  pipe  F,  so, 
during  a  fire,  when  many  streams  of  water  are  thrown  from  the 
same  main,  the  heavy  flow  lowers  the  available  pressure  at  the 
hose. 

143.  Gravity  Flow  of  Liquids. — In  the  last  section,  where  the 
flow  of  water  in  level  pipes  was  discussed,  it  was  shown  that  a 
pressure  difference  sufficient  to  overcome  the  friction  of  the  water 
on  the  pipes  is  always  necessary  to  maintain 
such  flow.     In  the  case  of  pipes  which  are  not 
level,  but  have  a  slight  slope,   such  as   tile 
drains  and  sewer  pipes,  friction  between  the 
water  and  the  pipe  is  overcome,  not  by  differ- 
ence in  pressure,  but  by  a  component  of  the 
weight  of  the  water  itself.     The  weight  of  a 
car  on  a  grade  may  be  resolved  into  two 
components,  one  of  which  is  parallel  to  the 
FIG.  95.  grade  and  therefore  urges  the  car  down  the 

grade  (Sec.  19).  Likewise,  the  weight  of  the 
water  in  the  tile  drain  may  be  resolved  into  two  components,  one 
of  which  is  parallel  to  the  drain  and  therefore  urges  the  water 
along  the  drain.  If  the  slope  of  the  drain  is  increased,  the  com- 
ponent parallel  to  the  drain  becomes  larger,  and  the  flow  becomes 
more  rapid.  The  other  component  which  is  perpendicular  to 
the  drain  does  not  interest  us  in  the  present  discussion. 

The  flowing  of  the  water  in  a  river  is  maintained  in  the  same 
way  as  in  a  tile  drain.  The  bed  of  the  river  has  a  certain  average 
slope  down  stream.  The  component  of  the  weight  of  the  water 
in  the  river  which  is  parallel  to  the  bed,  constitutes  the  driving 
force  that  overcomes  the  friction  on  the  shores  and  on  the  bot- 
tom. At  points  where  the  slope  is  great  this  force  is  great  and 
"rapids"  exist. 

Velocity  of  Efflux,  Torricclli's  Theorem. — As  the  water  in  A  (Fig. 
95),  of  depth  hi,  flows  from  the  orifice  B  it  acquires  a  velocity  v,  givea 
by  the  equation  v  =  \//clghi. 


PROPERTIES  OF  FLUIDS  IN  MOTION  197 

Proof:  As  M  pounds  of  water  pass  through  orifice  B,  the  water  level 
in  A  is  lowered  slightly,  and  the  potential  energy  of  A  is  reduced  by 
Mhi  foot-pounds  or  Mghi  foot-poundals.  The  kinetic  energy  of  the 
M  Ibs.  of  flowing  water  is  %Mv*  (Eqs.  50  and  51,  Sec.  75).  From  the 
conservation  of  energy  it  follows  that  this  kinetic  energy  must  be  equal 
to  the  potential  energy  lost  by  the  tank;  i,e.,  %Mvz  =  Mghi,  from  which 
we  have  vz  =  2gh}  OTV  =  \/2gh^.  From  Eq.  14,  Sec.  34,  we  see  that  \/2gh 
is  the  velocity  acquired  by  a  body  in  falling  from  rest  through  a  height 
h.  By  this  proof,  known  as  Torricelli's  theorem,  we  have  shown  that  the 
velocity  oifree  efflux  produced  by  a  given  head  h  is  equal  to  the  velocity 
of  free  fall  through  this  same  height  h.  If  a  pipe  were  connected  to  B  of 
a  length  such  as  to  require  a  friction  head  of  %hi  to  maintain  the  flow  in 
it,  then  the  velocity  head  would  be  f  hi,  and  the  velocity  of  flow  in  the 
pipe  would  be  \/2g  X  |Ai  or  that  acquired  by  a  body  in  falling  a  distance 


144.  The  Siphon.—  The  siphon,  which  is  a  U-shaped  tube  T 
(Fig.  96),  may  be  used  to  withdraw  water  or  other  liquids  from 
tanks,  etc.  If  a  siphon  is  filled  with  water  and  stoppered  and 
then  inverted  and  placed  in  a  vessel  of  water  A,  as  shown,  it 
will  be  found  that  the  water  flows  from  A  through  T  to  B. 
There  must  be  an  unbalanced  pressure  that  forces  this  water 
through  T.  This  pressure  may  be  readily  found. 

Imagine,  for  a  moment,  a  thin  film  to  be  stretched  across  the 
bore  of  the  tube  at  C.  The  pressure  tending  to  force  this  film 
to  the  right,  minus  the  pressure  tending  to  force  it  to  the  left 
is  evidently  the  unbalanced  pressure  which  causes  the  flow  in 
the  actual  case.  The  former  pressure  is  the  atmospheric  pres- 
sure, frequently  called  B  (from  barometer)  minus  hidg,  or 
B  —  hidg,  while  the  latter  is  B  —  h2dg.  The  unbalanced  pressure 
is,  therefore, 

Unbalanced  pressure  =  (B—  hidg)  —  (B  —  hzdg)  =  (h2  —  hi)  dg  =  h3dg 

From  this  equation  we  see  that  the  difference  of  pressure  is 
proportional  to  the  difference  in  level  (h3)  of  surfaces  Si  and  St. 
In  addition  to  this  difference  in  level,  the  factors  that  determine 
the  rate  of  flow  are  the  length  of  the  tube,  the  smoothness  and 
size  of  its  bore,  and  the  viscosity  of  the  liquid. 

Since  atmospheric  pressure  cannot  support  a  column  of  water 
which  is  more  than  34  ft.  in  height  (Sec.  136),  it  follows  that  hi 
(Fig.  96)  must  not  exceed  this  height  or  the  atmospheric  pressure 
on  Si  will  not  force  the  water  up  to  C,  and  the  siphon  will  fail  to 


198 


MECHANICS  AND  HEAT 


operate.  In  case  mercury  is  the  liquid  used,  hi  must  not  be  more 
than  29  or  30  inches.  If  made  greater  than  this,  a  vacuum  will 
be  formed  at  C  and  no  flow  will  take  place.  Since  a  partial 
vacuum  is  formed  at  C,  the  siphon  walls  must  not  easily  collapse. 

Observe  that  the  water  flows  from  point  a  to  point  6,  both 
points  being  at  the  same  pressure,  namely,  atmospheric  pressure. 
In  Sec.  143  it  was  shown  that  pressure  difference  is  not  the  only 
thing  which  may  maintain  a  steady  flow,  but  that  in  sloping 
pipes  a  component  of  the  weight  of  the  liquid  overcomes  the  fric- 
tion resistance.  In  vertical  pipes  the  full  weight  of  the  liquid 
maintains  the  flow.  Hence,  in  the  case  of 
the  siphon  we  may  consider  that  it  is  the  ex- 
cess weight  of  the  right  column  over  that  of 
the  left  which  provides  the  force  that  over- 
comes the  friction  between  the  flowing  column 
and  the  tube  T. 

145.  The  Suction  Pump. — The  common 
"  suction  pump  "  used  for  cisterns  and  shallow 
wells,  is  shown  in  Fig.  97  in  three  stages  of 
operation.  The  "cylinder"  C  is  open  at  the 
top  and  closed  at  the  bottom,  except  for  a 
valve  a  which  opens  upward.  Within  C  is  a 
snug-fitting  piston  P,  containing  a  valve  6  also 
opening  upward.  D  is  a  pipe  extending  below 
the  surface  of  the  water.  As  P  is  forced  down- 
ward by  means  of  the  piston  rod  R  attached  to 
the  pump  handle,  valve  a  closes,  and  as  soon  as  the  air  in  E  is 
sufficiently  compressed  it  lifts  the  valve  6  and  escapes  (left 
sketch).  As  the  piston  rises  again,  6  closes,  and  the  remnant  of 
air  in  E  expands  to  fill  the  greater  volume,  thereby  having  its 
pressure  reduced  (according  to  Boyle's  Law)  below  one  atmos- 
phere. The  pressure  of  the  air  in  D  is,  of  course,  one  atmosphere. 
Hence  the  pressure  above  the  valve  a  is  less  than  the  pressure 
below  it,  causing  it  to  rise  and  admit  some  air  into  E  from  D. 
As  air  is  thus  withdrawn  from  D  the  pressure  of  the  remaining 
air  is  reduced  to  below  atmospheric  pressure,  consequently  the 
water  in  the  cistern,  which  is  exposed  to  full  atmospheric  pressure, 
is  forced  up  into  the  tube  D  (middle  sketch,  Fig.  97).  Another 
stroke  of  the  piston  still  further  reduces  the  air  pressure  in  E  and 
D,  and  the  water  is  forced  higher,  until  it  finally  passes  through 
valve  a  into  the  cylinder.  As  P  descends,  valve  a  closes,  and  the 


FIG.  96. 


PROPERTIES  OF  FLUIDS  IN  MOTION 


199 


water  in  the  cylinder  is  forced  through  6,  and  finally,  as  P  again 
rises  fright  sketch),  it  is  forced  out  through  the  spout  d. 

Atmospheric  pressure  will  support  a  column  of  water  about  34 
ft.  in  height  (Sec.  136),  provided  the  space  above  the  column  is  a 
vacuum.  Hence  we  see  that  the  theoretical  limiting  vertical 
distance  from  the  cylinder  to  the  water  in  the  cistern,  or  well,  is 


R 
H 

—  ' 

- 

--P- 

:*: 

^ 

ti 

FIG.  97. 

34  ft.  Suppose  this  distance  to  be  40  ft.  Then,  even  if  a  per- 
fect vacuum  could  be  produced  in  E,  the  water  would  still  be  6  ft. 
below  the  cylinder.  In  practice,  the  cylinder  should  not  be 
more  than  20  or  25  ft.  above  the  water.  For  this  reason,  pumps 
for  deep  wells  have  the  cylinder  near  the  bottom,  the  piston  rod 
in  some  cases  being  several  hundred  feet  in  length. 


200 


MECHANICS  AND  HEAT 


146.  The  Force  Pump. — The  force  pump  is  used  when  it  is 
desired  to  pump  water  into  a  tank  which  is  at  a  higher  level  than 
the  pump.     The  pump  described  in  the  last  section  is  sometimes 
provided  with  a  tight-fitting  top  at  H  (right  sketch,  Fig.  97) 

having  a  hole  just  large  enough 
to  permit  the  piston  rod  R  to 
|  pass  through  it.     By  connecting 

spout  d  with  a  hose,  the  pump 
may  then  be  operated  as  a  force 
pump. 

The  other  type  of  force  pump 
(Fig.  98)"lifts"the  water  from  the 
well  on  the  upstroke,  and  forces 
it  up  to  the  tank  on  the  down- 
stroke,  thus  making  it  run  more 
evenly,  since  both  strokes  are 
working  strokes.  In  this  type 
the  piston  has  no  valve.  As 
the  piston  P  rises,  valve  b  closes 
and  valve  a  opens,  permitting 
water  to  enter  the  cylinder.  As 
P  descends,  a  closes  and  6  opens, 
and  the  water  is  forced  up  into 
the  tank.  During  the  down- 
stroke  of  P,  some  of  the  water 
rushes  into  the  air  chamber  A 
and  further  compresses  the  en- 
closed air.  During  the  upstroke 
(valve  6  being  then  closed)  this 
air  expands  slightly  and  expels 
some  water.  Thus,  by  the  use 
of  the  air  chamber,  the  flow  of 
water  from  the  discharge  pipe 
is  made  more  nearly  uniform. 
The  fact  that  the  descending 
piston  may  force  some  water  into 

A  instead  of  suddenly  setting  into  motion  the  entire  column 
of  water  in  the  vertical  pipe,  causes  the  pump  to  run  more 
smoothly. 

147.  The  Mechanical  Air  Pump. — The  mechanical  air  pump 
operates  in  exactly  the  same  way  as  the  suction  pump  (Fig.  97). 


D 


FIG.  98. 


PROPERTIES  OF  FLUIDS  IN  MOTION          201 

In  fact,  when  first  started,  the  suction  pump  withdraws  air  from 
D,  that  is,  it  acts  as  an  air  pump.  To  withdraw  the  air  from  an 
inclosure  (e.g.,  from  an  incandescent  lamp  bulb),  the  tube 
D  would  be  connected  to  the  bulb  instead  of  to  the  cistern.  The 
process  by  which  the  air  is  withdrawn  from  the  bulb  is  the  same 
as  that  by  which  it  is  withdrawn  from  D  (Sec.  145),  and  need  not 
be  redescribed  here.  As  exhaustion  proceeds,  the  air  pressure 
in  the  bulb  and  in  D  becomes  too  feeble  to  raise  the  valves. 
Hence  the  practical  air  pump  must  differ  from  the  suction  water 
pump  in  that  its  valves  are  operated  mechanically.  The  valves 
and  piston  must  also  fit  much  more  accurately  for  the  air  pump 
than  is  required  for  the  water  pump. 

The  upper  end  of  the  cylinder  of  an  air  pump  has  a  top  in 
which  is  a  small  hole  covered  by  the  outlet  valve.  If  a  pipe 
leads  from  this  valve  to  an  enclosed  vessel  the  air  will  be  forced 
into  the  vessel.  In  such  case  tube  D  would  simply  be  opened 
to  the  air,  and  the  pump  would  then  be  called  an  Air  Compressor. 
Such  air  compressors  are  used '  to  furnish  the  compressed  air 
for  operating  pneumatic  drills,  the  air-brakes  on  trains,  and  for 
many  other  purposes.  It  will  be  observed  that  such  an  air 
pump,  like  practically  all  pumps  (see  Fig.  98),  produces  suction 
at  the  entrance  and  pressure  at  the  exit. 

Let  us  further  consider  the  process  of  pumping  air  from  a  bulb 
connected  to  the  tube  D.  Assuming  perfect  action  of  the  piston  and 
valve,  and  assuming  that  the  volume  of  the  cylinder  is  equal  to  the 
combined  volume  of  the  bulb  and  D,  we  see  that  the  first  stroke  would 
reduce  the  pressure  in  bulb  and  D  to  1/2  atmosphere.  For  as  P  rises 
to  the  top  of  the  cylinder,  the  air  in  bulb  and  D  expands  to  double  its 
former  volume,  and  hence  the  pressure,  in  accordance  with  Boyle's  Law, 
decreases  to  1/2  its  former  value.  A  second  upstroke  reduces  the  pres- 
sure in  bulb  and  D  to  1/4  atmosphere,  a  third  to  1/8  atmosphere,  a 
fourth  to  1/16  atmosphere,  etc.  Observe  that  each  stroke  removes  only 
1/2  of  the  air  then  remaining  in  the  bulb. 

The  Geryk  Pump. — In  the  ordinary  mechanical  air  pump  there  is  a 
certain  amount  of  unavoidable  clearance  between  the  piston  and  the 
end  of  the  cylinder.  The  air  which  always  remains  in  this  clearance 
space  at  the  end  of  the  stroke,  expands  as  the  piston  moves  away,  and 
produces  a  back  pressure  which  finally  prevents  the  further  removal  of 
air  from  the  intake  tube,  and  therefore  lowers  the  efficiency  of  the  pump. 
In  the  Geryk  pump,  air  is  eliminated  from  the  clearance  space  by  the 
use  of  a  thin  layer  of  oil  both  above  and  below  the  piston. 

148.  The  Sprengel  Mercury  Pump. — The  Sprengel  pump  exhausts 


202 


MECHANICS  AND  HEAT 


very  slowly,  but  by  its  use  a  very  much  better  vacuum  may  be  obtained 
than  with  the  ordinary  mechanical  air  pump.  It  consists  essentially 
of  a  vertical  glass  tube  A  (Fig.  99)  about  one  meter  in  length  and  of 
rather  small  bore,  terminating  above  in  a  funnel  B  into  which  mercury 
may  be  poured.  A  short  distance  below  the  funnel  a  side  tube  leads  Irom 
the  vertical  tube  to  the  vessel  C  to  be  exhausted.  As  the  mercury  drops, 
one  after  another,  pass  down  through  the  vertical  tube  into  the  open 
dish  below,  each  drop  acts  as  a  little  piston  and  pushes  ahead  of  it  a 
small  portion  of  air  that  has  entered  from  the  side  tube.  Thus  any 
vessel  connected  with  this  side  tube  is  exhausted. 

Obviously,  to  obtain  a  good  vacuum,  the  aggregate  length  of  the  little 
mercury  pistons  below  the  side  tube  must  be  greater 
than  the  barometric  height,  or  the  atmospheric 
pressure  would  prevent  their  descent.  The  funnel 
must  always  contain  some  mercury,  or  air  will  en- 
ter and  destroy  the  vacuum.  A  valve  at  a  is  ad- 
justed to  permit  but  a  slow  flow  of  mercury, 
thereby  causing  the  column  to  break  into  pistons. 

149.  The  Windmill  and  the  Electric  Fan.— 
The  common  Windmill  consists  of  a  wheel 
I A     whose  axis  lies  in  the  direction  of  the  wind  and 


FIG.  99. 


FIG.  100. 


is  therefore  free  to  rotate  at  right  angles  to  the  direction  of  the 
wind.  This  wheel  carries  radial  vanes  which  are  set  obliquely 
to  the  wind  and  hence  to  the  axis  of  the  wheel.  In  Fig.  100, 
AB  is  an  end  view  of  a  vane  which  extends  toward  the  reader 
from  the  axis  (CD)  of  the  windmill  wheel.  From  analogy  to 
the  problem  of  the  sailboat  (Sec.  20),  we  see  at  once  that  the 
reaction  of  the  wind  w  against  the  vane  AB  gives  rise  to  a  thrust 
F  normal  to  the  vane.  This  force  may  be  resolved  into  the  two 
components  F\  and  F2.  Fz  gives  only  a  useless  end  thrust  on 
the  wheel  axle;  while  FI  gives  the  useful  force  which  drives  the 
vane  in  the  direction  FI.  When  the  vane  comes  to  a  position 
directly  below  the  axis  of  the  wheel,  FI  is  directed  away  from 


PROPERTIES  OF  FLUIDS  IN  MOTION  203 

the  reader.  Thus  in  these  two  positions,  and  indeed  in  all  other 
positions  of  the  vane,  FI  gives  rise  to  a  clockwise  torque  as 
viewed  from  a  point  from  which  the  wind  is  coming.  Every 
vane  gives  rise  to  a  similar,  constant,  clockwise  torque. 

The  ordinary  electric  fan  is  very  similar  to  the  windmill  in 
its  operation,  except  that  the  process  is  reversed.  In  the  case 
of"  the  windmill,  the  wind  drives  the  wheel  and  generates  the 
power;  while  with  the  fan,  the  electric  motor  furnishes  the  power 
to  drive  the  fan  and  produce  the  "wind."  In  the  former,  the 
reaction  between  the  vane  and  the  air  pushes  the  vane;  while  in^. 
the  latter  it  pushes  the  air. 

150.  Rotary  Blowers  and  Rotary  Pumps. — Blowers  are  used 
for  a  great  variety  of  purposes.  Important  among  these  are 
the  ventilation  of  mines;  the  production  of  the  forced  draft  for 
forges  and  smelting  furnaces;  the  production  of  the  "wind"  for 
fanning  mills  and  the  wind-stackers  on  threshing  mchines;  and 
for  the  production  of  "suction,"  as  in  the  case  of  tubes  that  suck 
up  shavings  from  wood-working  machinery,  foul  gases  from 
chemical  operations,  and  dust  as  in  the  Vacuum  Cleaner.  All 
blowers  may  be  considered  to  be  pumps,  and  like  all  pumps^ 
they  are  capable  of  exhausting  on  one  side  and  compressing  on 
the  other,  as  pointed  out  in  Sec.  147.  Hence  the  above  blowers 
that  produce  the  "wind"  do  not  differ  essentially  from  those 
that  produce  "suction."  Indeed  many  of  the  large  ventilating 
fans  used  in  mines  may  be  quickly  changed  so  as  to  force  the 
air  into  the  shaft,  instead  of  "drawing"  it  from  the  shaft. 

Blowers  commonly  produce  a  pressure  of  one  pound  per 
square  inch  or  less  (i.e.,  a  difference  from  atmospheric  pressure 
of  1  Ib.  per  sq.  in.),  although  the  so-called  "positive"  blowers 
may  produce  eight  or  ten  pounds  per  square  inch.  For  the 
production  of  highly  compressed  air,  such  as  used  in  the  air- 
brakes on  trains,  the  piston  air  pump  is  used  (see  Air 
Compressor,  Sec.  147). 

Rotary  Blowers. — Rotary  blowers  are  of  two  kinds,  disc  blowers 
and  centrifugal  blowers.  A  disc  fan  has  blades  which  are  radial 
and  set  obliquely  to  its  axis  of  rotation;  while  the  fan  proper 
has  its  blades  parallel  to  its  axis  and  usually  about  radial  (like 
the  blades  of  a  steamboat  paddle  wheel).  The  common  electric 
fan  is  of  the  former  type,  and  the  fan  used  in  the  fanning  mill  is 
of  the  latter  type.  If  a  disc  fan  is  placed  at  the  center  of  a  tube 
with  its  axis  parallel  to  the  tube,  it  will,  when  revolved,  force  a 


204 


MECHANICS  AND  HEAT 


stream  of  air  through  the  tube.  The  diameter  of  the  fan  should 
be  merely  enough  less  than  that  of  the  tube  to  insure  "  clearance." 
Such  a  blower  will  develop  at  the  intake  end  of  the  tube  a  slight 
suction  and  at  the  other  end  a  slight  pressure.  This  type  is 
widely  used  for  ventilation  purposes. 

The  essential  difference  between  the  Turbine  Pump  and  the 
blower  just  described  is  that  the  fan  is  stronger  and  propels"  a 
stream  of  water  instead  of  a  stream  of  air.  The  turbine  pump  is 
useful  in  forcing  a  large  quantity  of  water  up  a  slight  grade  for 
a  short  distance.  It  is  not  a  high  pressure  pump. 

The  Screw  Propeller,  universally  used  on  ocean  steamships  and 
also  used  on  gasoline  launches,  is  essentially  a  turbine  pump. 
The  propeller  forces  a  stream  of  water  backward  and  the  reacting 
thrust  forces  the  ship  forward. 

The  Centrifugal  Blower  is  similar  in  its  action  to  the  centrifugal 
pump  described  below. 

The  Centrifugal  Pump. — One  type  of  centrifugal  pump,  shown 
in  section  in  Fig.  101,  consists  of  a  wheel  W,  an  intake  pipe  A 


FIG.  101. 

which  brings  water  to  the  center  of  the  wheel,  and  an  outflow 
pipe  B,  which  conveys  the  water  from  the  periphery  of  the  wheel. 
The  direction  of  flow  of  the  water  at  various  points  is  indicated 
by  the  arrows.  By  means  of  an  electric  motor  or  other  source 
of  power,  the  wheel  is  rotated  in  the  direction  of  arrow  a,  and  the 
centrifugal  force  thereby  developed  causes  the  water  to  flow 
radially  outward  through  the  curved  passages  in  the  wheel  as 
indicated  by  the  arrows.  In  this  way,  it  is  feasible  to  produce 
a  pressure  of  20  or  30  Ibs.  per  sq.  in.  in  the  space  C,  which  is 
sufficient  pressure  to  force  water  to  a  vertical  height  of  50  or  60 
ft.  in  a  pipe  connected  with  B. 


PROPERTIES  OF  FLUIDS  IN  MOTION  205 

If  it  is  desired  to  raise  water  to  a  greater  height  than  this, 
several  pumps  can  be  used  in  "  series."  In  such  a  series  arrange- 
ment, the  lowest  pump  would  force  water  through  outlet  pipe  B 
to  the  intake  of  a  similar  pump,  say  50  ft.  above.  This  second 
pump  would  force  the  water  on  to  the  next  above,  and  so  on. 

151.  The  Turbine  Water  Wheel.— The  Turbine  Water  Wheel 
operates  on  the  same  general  principle  as  the  windmill;  a  stream 
of  water  driving  the  former,  a  stream  of  air  the  latter.     Since 
water  is  much  more  dense  than  air,  turbine  water  wheels  develop 
a  great  deal  more  power  than  windmills  of  the  same  size.     At 
the  Niagara  Falls  power  plant,  water  under  about  150  ft.  verti- 
cal head  rushes  into  the  great  turbines,  each  of  which  develops 
5000  H.P.     Turbines  of  10,000  H.P.  each  are  used  in  the  power 
plant  at  Keokuk,  Iowa. 

There  are  several  kinds  of  water  turbines.  In  the  "Axial  Flow 
Turbines"  in  which  the  water  flows  parallel  to  the  axis,  the  ac- 
tion of  the  windmill  is  practically  duplicated;  so  that  Fig.  100  and 
the  accompanying  discussion  would  apply  to  a  vane  of  such  a 
turbine,  provided  w  were  to  represent  moving  water  instead  of 
moving  air.  In  the  "Radial  Flow  Turbines"  the  water  flows  in  a 
general  radial  direction  either  toward  or  away  from  the  axis. 

If  water  under  considerable  pressure  is  forced  in  at  pipe  A 
(Fig.  101)  through  wheel  W,  and  out  at  pipe  B,  it  will  drive  W 
in  the  direction  of  arrow  a.  For,  as  the  water  flows  outward 
through  the  curved  radial  passages,  it  would,  by  virtue  of  its 
inertia,  produce  a  thrust  against  the  concave  wall  of  the  passage. 
This  thrust  would  clearly  produce  a  positive  (left-handed) 
torque.  Under  these  circumstances,  the  wheel  would  develop 
power,  and  would  be  called  a  radial  flow  turbine  water  wheel. 

The  Steam  Turbine,  used  to  obtain  power  from  steam,  is  similar 
to  the  water  turbine  in  principle,  but  greatly  differs  from  it  in 
detail.  The  development  of  the  light,  high  power,  high  efficiency 
steam  turbine  is  among  the  comparatively  recent  achievements 
of  steam  engineering.  The  steam  turbine  is  further  considered 
in  Sec.  235. 

152.  Pascal's  Law. — The  fact  that  liquids  confined  in  tubes,, 
etc.,  transmit  pressure  applied  at  one  point  to  all  points,  has 
already  been  pointed  out  (Sec.  114).     This  is  known  as  Pascal's 
Law.     Pascal's  Law  holds  with  regard  to  gases  as  well  as  liquids. 

This  law  has  many  important  applications,  among  which  are 
the  transmission  of  pressure  by  means  of  the  water  mains  to  all 


206  MECHANICS  AND  HEAT 

parts  of  the  city,  and  the  operation  of  the  hydraulic  press  and 
the  hydraulic  elevator. 

153.  The  Hydraulic  Press.— The  hydraulic  press  (Fig.  102) 
is  a  convenient  device  for  securing  a  very  great  force,  such  as 
required  for  example  in  the  process  of  baling  cotton.     It  con- 
sists of  a  large  piston  or  plunger  P,  fitting  accurately  into  a  hole 
in  the  top  of  a  strong  cylindrical  vessel  B.     As  water  is  forced 
into  B  by  means  of  a  force  pump  connected  with  pipe  D,  the 
plunger  P   rises.     As  P  rises,    the   platform  C  compresses  the 
cotton  which  occupies  the  space  A.     By  opening  a  valve  E,  the 
water  is  permitted  to  escape  and  P  descends. 

In  accordance  with  Pascal's  Law,  the  pressure  developed  by 
the  force  pump  is  transmitted  through  D  to 
the  plunger  P.  It  will  be  observed  that  since 
the  pressure  on  the  curved  surface  of  the 
plunger  is  perpendicular  to  that  surface  (Sec. 
114),  it  will  have  no  tendency  to  lift  the 
plunger.  The  lifting  force  is  pAi,  in  which 
AI  is  the  area  of  the  bottom  end  of  the 
plunger,  and  p,  the  water  pressure. 

If  the  area  of  the  piston  of  the  force  pump  is 
A  2,  then,  since  the  pressure  below  this  piston 
FIG.  102.  is  practically  the  same  as  that  acting  upon 

plunger  P,  it  follows  that  the  lift  exerted  by  P 
will  be  greater  than  the  downward  push  upon  the  piston  of  the 
force  pump  in  the  ratio  of  AI  to  Az.  In  other  words,  the  theo- 
retical mechanical  advantage  is  Ai-i- A%. 

Instead  of  using  a  force  pump,  the  pipe  D  may  be  connected 
to  the  city  water  system.  If  this  pressure  is  100  Ibs.  per  in.2, 
and  if  ^li  =  100  in.2,  then  P  will  exert  a  force  of  10,000  Ibs.  or 
5  tons.  With  some  steel  forging  presses  a  force  of  several 
thousand  tons  is  obtained. 

154.  The  Hydraulic  Elevator. — The  simplest  form  of  hydraulic 
elevator,  known  as  the  direct-connected  or  direct-lift  type,  is 
the  same  in  construction  and  operation  as  the  hydraulic  press 
.(Fig.  102),  except  that  the  plunger  is  longer.     If  the  elevator, 
built  on  platform  C,  is  to  have  a  vertical  travel  of  30  ft.,  then  the 
plunger  P  must  be  at  least  30  ft.  in  length. 

In  another  type  of  hydraulic  elevator,  the  plunger  and  con- 
taining cylinder  lie  in  a  horizontal  position  in  the  basement  of 
the  building.  The  plunger  is  then  connected  with  the  elevator 


PROPERTIES  OF  FLUIDS  IN  MOTION 


207 


by  means  of  a  system  of  gears  or  pulleys  and  cables  in  such  a 
way  that  the  elevator  travels  much  farther,  and  hence  also  much 
faster  than  the  plunger.  This  type  is  much  better  than  the 
direct-connected  type  for  operating  elevators  in  high  buildings. 
In  both  types  the  valves  that  regulate  the  flow  of  water  to  and 
from  the  cylinder  are  controlled  from  the  elevator. 

155.  The  Hydraulic  Ram. — The  hydraulic  ram  (shown  in 
Fig.  103)  depends  for  its  action  upon  the  high  pressure  developed 
when  a  moving  stream  of  water  confined  in  a  tube  is  suddenly 
stopped.  It  is  used  to  raise  a  small  percentage  of  the  water 
from  a  spring  or  other  source  to  a  considerable  height. 

The  valve  C  is  heavy  enough  so  that  the  water  pressure  lidg 
(see  figure)  is  not  quite  sufficient  to  keep  it  closed.  As  it  sinks 
slightly,  the  water  flows  rapidly  past  above  it;  while  at  the 
same  time  the  water  below  it  is  practically  still.  In  the  next 


FIG.  103. 

section  it  will  be  shown  that  the  pressure  on  a  fluid  becomes 
lower  the  faster  it  moves;  accordingly,  the  pressure  above  the 
valve  is  less  than  the  pressure  below,  and  the  valve  rises  and 
closes.  The  closing  of  this  valve  suddenly  checks  the  motion 
of  the  water  in  pipe  B.  But  suddenly  stopping  any  body  in 
motion  requires  a  relatively  large  accelerating  force  (negative), 
hence  here  a  considerable  pressure  is  developed  in  pipe  B.  This 
"instantaneous,"  or  better  brief,  pressure  opens  the  valve  D 
and  forces  some  water  into  the  air  chamber  E  and  also  into 
pipe  F.  Valve  C  now  sinks,  and  the  operation  is  repeated,  forc- 
ing still  more  water  in  E,  until  finally  the  water  is  forced  through 
F  into  a  supply  tank  which  is  on  higher  ground  than  the  source 
A.  The  action  of  the  air  chamber  is  explained  in  Sec.  146. 

If  the  hydraulic  ram  had  an  efficiency  of  100  per  cent.,  then, 
from  the  conservation  of  energy,  we  see  that  it  would  raise  1/n 


208 


MECHANICS  AND  HEAT 


of  the  total  amount  of  water  to  a  height  nh.  Its  efficiency,  how- 
ever, is  only  about  60  per  cent.;  hence  it  will  force  l/n  of  the 
total  water  used  to  a  height  0.6  nh. 

156.  Diminution  of  Pressure  in  Regions  of  High  Velocity. — If 
a  stream  of  air  is  forced  rapidly  through  the  tube  A  (Fig.  104), 
it  will  be  found  that  the  pressure  at  the  restricted  portion  B  is 
less  than  elsewhere,  as  at  C  or  D.  If  the  end  D  is  short  and 
open  to  the  air,  manometer  F  will  indicate  that  D  is  practically 
at  atmospheric  pressure.  The  pressure  at  C  will  be  slightly 
above  atmospheric  pressure,  as  indicated  by  manometer  E. 
That  the  pressure  at  B  is  less  than  one  atmosphere,  and  hence 
less  than  at  either  C  or  D,  is  evidenced  by  the  fact  that  the  liquid 
stands  higher  in  tube  0  than  in  the  vessel  H. 


H 


FIG.  104. 


That  the  pressure  at  B  should  be  less  than  at  C  or  D  is  ex- 
plained as  follows:  Since  the  tube  has  a  smaller  cross  section 
at  B  than  at  C  or  D,  it  is  evident  that  the  air  must  have  a  higher 
velocity  at  B  than  at  the  other  two  points,  as  indicated  in  the 
figure  by  the  difference  in  the  length  of  the  arrows.  As  a  particle 
of  air  moves  from  C  to  B  its  velocity,  then,  increases.  To  cause 
this  increase  in  velocity  requires  an  accelerating  force.  Conse- 
quently the  pressure  behind  this  particle  tending  to  force  it  to 
the  right  must  be  greater  than  the  pressure  in  front  of  it,  tending 
to  force  it  to  the  left.  As  the  particle  moves  from  B  to  D  it 
slows  down,  showing  that  the  backward  pressure  upon  it  must 
be  greater  than  the  forward  pressure.  Thus  B  is  a"  region  of 
lower  pressure  than  either  C  or  D  simply  because  it  is  a  region 
of  higher  velocity.  The  reduction  in  pressure  at  B  is  explained 


PROPERTIES  OF  FLUIDS  IN  MOTION 


209 


by  means  of  Bernoulli's  theorem  under  "Venturi  Water  Meter" 
(see  below). 

The  Atomizer. — If  the  air  rushes  through  B  still  more  rapidly, 
the  pressure  will  be  sufficiently  reduced  so  that  the  liquid  will 
be  " drawn"  up  from  vessel  H  and  thrown  out  at  I  as  a  fine  spray. 
The  tube  then  becomes  an  atomizer. 

The  Aspirator  or  Filter  Pump. — A  similar  reduction  in  pressure 
occurs  at  B  if  water  flows  rapidly  through  the  tube.  Thus,  if 
the  tube  is  attached  in  a  vertical  position  to  a  faucet,  the  water 
rushing  through  B  produces  a  low  pressure  and  consequently 
"suction,"  so  that  if  a  vessel  is  connected  with  G  the  air  is 
withdrawn  from  it,  producing  a  partial  vacuum.  Under  these 
circumstances  the  tube  acts  as  a  filter  pump  or  aspirator. 

The  "forced  draft"  of  locomotives  is  produced  by  a  jet  of 
steam  directed  upward  in  the  smoke  stack. 

The  Jet  Pump. — If  a  stream  of  water  from  a  hydrant  is  directed 
through  B,  a  tube  connected  with  G  may  be  employed  to  "draw" 


FIG.  105. 

water  from  a  cistern  or  flooded  basement.  Such  an  arrangement 
is  a  jet  pump.  In  Fig.  105  a  jet  pump  is  shown  pumping  water 
from  a  basement  B  into  the  street  gutter  /.  Pipe  A  is  con- 
nected to  the  hydrant. 

Bernoulli's  Theorem. — Bernoulli's  Theorem,  first  enunciated 
in  1738  by  John  Bernoulli,  is  of  fundamental  importance  to  some 
phases  of  the  study  of  hydraulics.  We  shall  develop  this  theorem 
from  a  discussion  of  Fig.  95. 

Let  water  flow  into  A  at  the  top  as  rapidly  as  it  flows  out  at 
B,  thus  maintaining  a  constant  water  level.  Let  us  next  con- 
sider the  energy  possessed  by  a  given  volume  V  of  water  in  the 


210  MECHANICS  AND  HEAT 

different  stages  of  its  passage  from  the  surface  S  to  the  out- 
flowing stream  at  B.  Its  energy  (potential  energy  Ep)  when 
at  S  is  Mghi  CEq.  50,  Sec.  75),  and,  since  M=Vd  (volume  times 
density,  Sec.  101),  we  have 


As  this  given  volume  reaches  point  6  at  a  slight  distance  h  above 
the  orifice,  it  has  potential  energy  Mgh,  or  Vdgh,  and,  since  it 
now  has  appreciable  velocity  v,  it  has  kinetic  energy  %Mv*, 
or  %Vdv2.  In  addition  to  this  it  has  potential  energy,  because 
of  the  pressure  (p)  exerted  upon  it  by  the  water  above,  which 
energy,  we  shall  presently  prove,  is  pV.  Consequently,  its  total 
energy  when  at  b  is 

E=Vdgh+pV+$Vdv*  (73) 

Eq.  73  is  the  mathmetical  statement  of  Bernoulli's  theorem. 
If  C.G.S.  units  are  used  throughout  (i.e.,  if  V  is  given  in  cm.3, 
p  in  dynes  per  cm.2,  v  in  cm.  per  sec.,  etc.),  then  E  will  be  the 
energy  in  ergs.  If  the  volume  chosen  is  unity,  the  equation  re- 
duces to  E  =  dgh-\-p+%dv2,  a  form  frequently  given. 

Observe  that  when  the  volume  V  is  at  S,  p  and  v  are  zero,  hence 
E=  Vdghi  as  already  shown;  while  when  this  volume  reaches  the 
flowing  stream,  p  and  h  are  zero,  hence  E  =  %Vdv2  (i.e.,  \Mv*}. 
From  the  law  of  the  conservation  of  energy  we  know  that  these 
two  amounts  of  energy  must  be  equal,  i.e.,  Vdghi  =  %Vdv*, 
which  reduces  to  v=  '\l2ghi,  an  equation  already  deduced  (Sec. 
143)  from  slightly  different  considerations.  When  the  volume 
is  half  way  down  in  vessel  A,  Vdgh  =  pV,  and  the  third  term 
\Vdvz  is  practically  zero,  since  v  at  this  point  has  a  small  value. 
It  should  be  observed  that  when  the  volume  under  consideration 
is  below  the  surface,  then  the  height  measured  from  the  volume 
up  to  the  surface  determines  the  pressure;  whereas  the  height 
measured  from  the  volume  down  to  the  orifice,  determines  the 
potential  energy  due  to  the  elevated  position.  Obviously  the 
energy  due  to  elevation  decreases  by  the  same  amount  that 
the  energy  due  to  pressure  increases,  and  vice  versa,  and  the 
sum  of  these  two  amounts  of  energy  is  constant  so  long  as  the 
velocity  v  (last  term  Eq.  73)  is  practically  zero. 

We  shall  now  prove  that  the  potential  energy  of  the  above 
volume  V,  when  subjected  to  a  pressure  p,  is  pV.  Let  the  volume 
V,  as  it  passes  out  at  B,  slowly  push  a  snug-fitting  piston  in  B  a 


PROPERTIES  OF  FLUIDS  IN  MOTION  211 

distance  di  such  that  diAi  =  V,  in  which  Ai  is  the  cross  section 
of  the  orifice.  The  work  done  by  the  volume  V  on  the  piston 
is  pAiXdi  (force  times  distance),  which  shows  that  the  potential 
energy  of  V  immediately  before  exit  was  pXAidi  or  pV. 

The  Venturi  Water  Meter.—  The  Venturi  water  meter,  used 
for  measuring  rate  of  flow,  differs  from  the  apparatus  sketched 
in  Fig.  104  in  that  the  medium  is  water  instead  of  air,  and  the 
pressure  is  measured  by  ordinary  pressure  gages  instead  of 
as  shown.  If  pipe  A  were  6  ft.  in  diameter  at  C,  it  would  taper 
in  a  distance  of  100  ft.  or  so  to  a  diameter  of  about  2  ft.  at  B. 

Let  the  pressure,  area  of  cross  section,  and  velocity  of  flow 
at  C  and  B,  respectively,  be  pe,  Ac,  vc,  and  pb,  Ab,  vb.  Now  the 
energy  of  a  given  volume  V  when  at  C  must  be  equal  to  its 
energy  when  at  B;  hence,  from  Eq.  73,  we  have 

Vdgh  +  PC  V  +  1  Vdvc*  =  Vdgh  +  pb  V  +  £  Vdvb* 
from  which  we  get 

».f)  (74) 


Since  in  unit  time  equal  volumes  must  pass  B  and  C,  we  have 


e 

•n-b 

Substituting  in  Eq.  74  this  value  of  vb  gives 


or   vb  =  -T^ve  (74a) 


If  the  pressure  is  reduced  to  poundals  per  square  foot,  the 
cross  section  to  square  feet,  and  if  the  density  of  the  water  is 
also  expressed  in  the  British  system  (i.e.,  62.4  Ibs.  per  cu.  ft.), 
then  vc  will  be  expressed  in  feet  per  second.  Multiplying  vc 
by  Ac  (in  square  feet)  gives,  for  the  rate  of  flow,  vcAc,  in  cubic 
feet  per  second. 

157.  The  Injector. — Injectors  are  used  for  forcing  water  into 
boilers  while  the  steam  pressure  is  on.  Their  operation  depends 
upon  the  decrease  of  pressure  produced  by  the  high  velocity  of 
a  jet  of  steam,  coupled  with  the  condensation  of  the  steam  in 
the  jet  by  contact  with  the  water  spray  brought  into  the  jet  by 
the  atomizer  action  (Sec.  156).  Some  of  the  commercial  forms 
of  the  injector  are  quite  complicated. 

The  injector  shown  diagrammatically  in  Fig.  106  is  compara- 


212  MECHANICS  AND  HEAT 

tively  simple.  If  valves  a  and  d  are  opened,  b  being  closed,  the 
steam  from  the  boiler  B  rushes  through  D,  E  and  e  and  out  at 
a  into  the  outside  air.  The  steam,  especially  at  the  restricted 
portion  E  of  the  tube,  has  a  very  high  velocity,  and  hence,  from 
Sec.  156,  we  see  that  a  low  pressure  exists  at  E.  The  pressure 
at  E  being  less  than  one  atmosphere,  the  atmospheric  pressure 
upon  the  water  in  the  tank  forces  water  up  through  the  pipe 
P  into  E,  where  it  passes  to  the  right  with  the  steam  which 
quickly  condenses.  This  stream  of  water,  due  to  its  momentum, 
raises  check  valve  b  and  passes  into  the  boiler  against  the  boiler 
pressure.  As  soon  as  the  flow  through  b  is  established,  valve  a 
should  be  closed.  In  many  injec- 
tors, the  suction  due  to  the  par- 
tial vacuum  at  e  automatically 
closes  a  check  valve  opening  down- 
ward at  a. 

It  should  be  pointed  out  that  in 
the  action  of  the  injector,  by  which 
steam  under  a  pressure  p  forces 
supply  water  (and  also  the  con- 
densed steam)  into  the  boiler  against 
this  same  steam  pressure  plus  a 
FIG.  106.  slight  water  pressure  (see  figure), 

there  is  no  violation  of  the  law  of 

the  conservation  of  energy  The  energy  involved  is  pressure 
times  volume  in  both  cases,  but  the  volume  of  water  forced  into 
the  boiler  in  a  given  time  is  much  less  than  the  volume  of  steam 
used  by  the  injector. 

158.  The  Ball  and  Jet— If  a  stream  of  air  B,  Fig.  107,  is 
directed  as  shown  against  a  light  ball  A,  e.g.,  a  ping  pong  ball 
or  tennis  ball,  the  ball  will  remain  in  the  air  and  rapidly  revolve 
in  the  direction  indicated. 

The  explanation  is  simple.  There  are  three  forces  acting 
upon  the  ball,  namely,  W,  F\,  and  F2,  as  shown.  The  force  FI 
arises  from  the  impact  of  the  stream  of  air  B.  The  force  Fz  is 
due  to  the  fact  that  the  air  pressure  at  a  is  less  than  at  b.  The 
pressure  at  6  is  one  atmosphere,  while  at  a  it  is  slightly  less  be- 
cause a  is  a  region  of  high  velocity.  W  represents  the  weight 
of  A.  If  it  is  desired  to  determine  the  magnitude  of  F\  and  Fz, 
the  magnitude  of  W  may  be  found  by  weighing  A,  and  then, 
since  the  ball  is  in  equilibrium,  these  three  forces  W,  FI,  and 


PROPERTIES  OF  FLUIDS  IN  MOTION 


213 


FIG.  107. 


Fz,  acting  upon  it  must  form  a  closed  triangle,  as  explained  in 
Sec.  18. 

Card  and  Spool. — If  a  circular  card,  having  a  pin  inserted 
through  the  center,  is  placed  below  a  spool  through  the  center  of 
which  a  rapid  stream  of  air  is  blown,  it  will  be  found  that  the  card 
will  be  supported  in  spite  of  the  downward  rush  of  air  upon  it 
which  might  be  expected  to  blow 
it  away.  The  air  above  the  card 
is  moving  rapidly  in  all  directions 
away  from  the  center;  consequent- 
ly the  region  between  the  spool  and 
card,  being  a  region  of  high  veloc- 
ity, is  also  a  region  of  low  pressure 
—lower,  in  fact,  than  the  pressure 
below  the  card.  This  difference  in 
pressure  will  not  only  support  the 
weight  of  the  card,  but  also  addi- 
tional weight. 

159.  The  Curving  of  a  Baseball. 
— The  principle  involved  in  the 
pitching  of  "in  curves,"  "out 

curves, "  etc.,  will  be  understood  from  a  discussion  of  Fig.  108. 
Let  A  represent  a  baseball  rotating  as  indicated,  and  moving  to 
the  right  with  a  velocity  v.  If  A  were  perfectly  frictionless,  the 
air  would  rush  past  it  equally  fast  above  and  below,  i.e.,  vv 
and  #2  would  be  equal.  (We  are  familiar  with  the  fact  that  a 
person  running  10  mi.  per  hr.  east  through  still  air,  faces  a  10  mi. 
per  hr.  breeze  apparently  going  west.)  If  the  surface  of  the  ball 

is  rough,  however,  it  will  be  evi- 
dent that  where  this  surface  is 
moving  in  the  direction  of  the 
rush  of  air  past  it,  as  on  the 
upper  side,  it  will  not  retard  that 
rush  so  much  as  if  it  were  mov- 
ing in  the  direction  opposite  to 
the  rush  of  air,  as  it  clearly  is 
The  air,  then,  rushes  more  read- 
ily, and  hence  more  rapidly,  past  the  upper  surface  than  past 
the  lower  surface  of  the  ball;  hence,  as  the  ball  moves  to  the 
right,  the  air  pressure  above  it  is  less  than  it  is  below,  and  an 
"up  curve"  results. 


FIG.  108. 


on  the  lower  side  of  the  ball. 


214  MECHANICS  AND  HEAT 

The  "drop  curve"  is  produced  by  causing  the  ball  to  rotate 
in  a  direction  opposite  to  that  shown;  while  the  "in  curve"  and 
"out  curve"  require  rotation  about  a  vertical  axis. 

A  lath  may  be  made  to  produce  a  very  pronounced  curve  by 
throwing  it  in  such  a  way  as  to  cause  it  to  rotate  rapidly  about 
its  longitudinal  axis,  the  length  of  the  lath  being  perpendicular 
to  its  path. 

PROBLEMS 

1.  A  force  pump,  having  a  3-ft.  handle  with  the  piston  rod  operated  by  a 
6-in.  "arm,"  (i.e.,  with  the  pivot  bolt  6  in.  from  one  end  of  the  handle),  and 
having  a  piston  head  2  in.  in  diameter,  is  used  to  pump  water  into  an  hydrau- 
lic press  whose  plunger  is  1.5  ft.  in  diameter.     What  force  will  a  100-lb. 
pull  on  the  end  of  the  pump  handle  exert  upon  the  plunger  of  the  press? 

2.  An  hydraulic  press  whose  plunger  is  2  ft.  in  diameter  is  operated  by  water 
at  a  pressure  of  600  Ibs.  per  sq.  in.     How  much  force  does  it  exert?     Express 
in  tons. 

3.  An  hydraulic  elevator  operated  by  water  under  a  pressure  of  100  Ibs. 
per  in.2  has  a  plunger  10  in.  in  diameter  and  weighs  2.5  tons.     How  much 
freight  can  it  carry? 

4.  If  /n  =  10  ft.,  and  h2  =  lS  ft.  (Fig.  96),  what  will  be  the  pressure  at  C 
(a)  if  the  left  end  of  the  siphon  is  stoppered?     (6)  If  the  right  end  is  stop- 
pered?   Assume  the  barometric  pressure  to  be  equal  to  that  due  to  34  ft. 
depth  of  water. 

6.  What  pressure  will  be  required  to  pump  water  from  a  river  into  a  tank 
on  a  hill  300  ft.  above  the  river,  if  20  per  cent,  of  the  total  pressure  is  needed 
to  overcome  friction  in  the  conveying  pipes? 

6.  How  long  will  it  take  a   10-H.P.  pump  (output  10  H.P.)  to  pump 
1000  cu.  ft.  of  water  into  the  tank  (Prob.  5)? 

7.  If  the  water  in  pipe  B  (Fig.  94)  flows  with  a  velocity  of  4  ft.  per  sec., 
what  will  be  the  value  of  &2?     Neglect  friction  head  in  the  portion  i  to  c" 
(Sees.  142  and  143). 

8.  What  would  be  the  limiting  (maximum)  distance  from  the  piston  to 
the  water  level  in  the  cistern  (Fig.  97)  at  such  an  altitude  that  the  baro- 
metric height  is  20  in.? 


PART  III 
HEAT 


CHAPTER  XIII 
THERMOMETRY  AND  EXPANSION 

160.  The  Nature  of  Heat. — As  was  pointed  out  in  the  study 
of  Mechanics,  a  portion  of  the  power  applied  to  any  machine 
is  used  in  overcoming  friction.  It  is  a  matter  of  everyday  ob- 
servation that  friction  develops  heat.  It  follows,  then,  that 
mechanical  energy  may  be  changed  to  heat.  In  the  case  of  the 
steam  engine  or  the  gas  engine  the  ability  to  do  work,  that  is 
to  run  the  machinery,  ceases  when  the  heat  supply  is  withdrawn. 
Therefore  heat  is  transformed  into  mechanical  energy  by  these 
engines,  which  on  this  account  are  sometimes  called  heat  engines. 

Heat,  then,  is  a  form  of  energy,  a  body  when  hot  possessing 
more  energy  than  when  cold.  Cold,  it  may  be  remarked,  is  not 
a  physical  quantity  but  merely  the  comparative  absence  of  heat, 
just  as  darkness  is  absence  of  light.  The  heat  energy  of  a  body 
is  supposed  to  be  due  to  a  very  rapid  vibration  of  the  molecules 
of  the  body.  As  a  body  is  heated  to  a  higher  temperature, 
these  vibrations  become  more  violent. 

It  has  been  proved  experimentally,  practically  beyond  ques- 
tion, that  both  radiant  heat  and  light  consist  in  waves  in  the 
transmitting  medium  (ether).  To  produce  a  wave  motion  in 
any  medium  requires  a  vibrating  body.  As  a  body,  for  example 
a  piece  of  iron,  becomes  hotter  and  hotter  it  radiates  more  heat 
and  light.  Hence,  since  the  iron  does  not  vibrate  as  a  whole, 
the  logical  inference  is  that  the  radiant  heat  and  light  are  pro- 
duced by  the  vibrations  of  its  molecular  or  atomic  particles. 

Until  about  one  hundred  years  ago  heat  was  supposed  to  be 
a  substance,  devoid  of  weight  or  mass,  called  Caloric,  which, 
when  added  to  a  body  caused  it  to  become  hotter,  and  when 
withdrawn  from  a  body  left  the  body  colder.  In  1798,  Count 
Rumford  showed  that  an  almost  unlimited  amount  of  heat  could 
be  taken  from  a  cannon  by  boring  it  with  a  dull  drill.  The  heat 
was  produced,  of  course,  by  friction.  In  the  process  a  very  small 
amount  of  metal  was  removed.  As  the  drilling  proceeded  and 
more  "caloric"  was  taken  from  the  cannon,  it  actually  became 
217 


218  MECHANICS  AND  HEAT 

hotter  instead  of  colder  as  the  caloric  theory  required.  Further- 
more, the  amount  of  heat  developed  seemed  to  depend  upon 
the  amount  of  work  done  in  turning  the  drill.  The  result  was 
the  complete  overthrow  of  the  caloric  theory. 

In  1843,  Joule  showed  by  experiment  that  if  772  ft.-lbs.  of 
work  were  used  in  stirring  1  Ib.  of  water,  its  temperature  would 
be  raised  1°  F.  This  experiment  showed  beyond  question  that 
heat  is  a  form  of  energy,  and  that  it  can  be  measured  in  terms 
of  work  units.  Later  determinations  have  given  778  ft.-lbs. 
as  the  work  necessary  to  raise  the  temperature  of  1  Ib.  of  water 
1°  F.  The  amount  of  heat  required  to  warm  1  Ib.  of  water  1°  F. 
is  called  the  British  Thermal  Unit  (B.T.U.);  so  that  1  B.T.U.  = 
778  ft.-lbs. 

161.  Sources  of  Heat. — As  already  stated,  Friction  is  one 
source  of  heat.  Rubbing  the  hands  together  produces  noticeable 
warmth.  Shafts  become  quite  hot  if  not  properly  oiled.  Primi- 
tive man  lighted  his  fires  by  vigorously  rubbing  two  pieces  of 
wood  together.  The  shower  of  sparks  from  a  steel  tool  held 
against  a  rapidly  revolving  emery  wheel,  and  the  train  of  sparks 
left  by  a  meteor  or  shooting  star,  show  that  high  temperatures 
may  be  produced  by  friction.  In  the  latter  case,  the  friction 
between  the  small  piece  of  rock  forming  the  meteor,  and  the 
air  through  which  it  rushes  at  a  tremendous  velocity,  develops, 
as  a  rule,  sufficient  heat  to  burn  it  up  in  less  than  a  second. 

Chemical  Energy. — Chemical  energy  is  an  important  source 
of  heat.  The  chemical  energy  of  combination  of  the  oxygen  of 
the  air  with  the  carbon  and  hydrocarbons  (compounds  of  car- 
bon and  hydrogen)  of  coal  or  wood,  is  the  source  of  heat  when 
these  substances  are  "burned,"  .that  is,  oxidized.  In  almost 
every  chemical  reaction  in  which  new  compounds  are  formed, 
heat  is  produced. 

The  Main  Source  of  heat  is  the  Sun.  The  rate  of  flow  of  heat 
energy  in  the  sun's  rays  amounts  to  about  1/4  H.P.  for  every 
square  foot  of  surface  at  right  angles  to  the  rays.  Upon  a  high 
mountain  this  amount  is  greater,  since  the  strata  of  the  air  below 
the  mountain  peak  absorb  from  10  to  20  per  cent,  of  the  energy 
of  the  sun's  rays  before  they  reach  the  earth.  On  the  basis  of 
1/4  H.P.  per  sq.  ft.,  the  total  power  received  by  the  earth  from 
the  sun  is  easily  shown  to  be  about  350  million  million  H.P. 
This  enormous  amount  of  power  is  only  about  1/2,000,000,000 
part  of  the  total  power  given  out  by  the  sun  in  all  directions. 


THERMOMETRY  AND  EXPANSION  219 

Obviously  a  surface  receives  more  heat  if  the  sun's  rays  strike 
it  normally  (position  AB,  Fig.  109)  than  if  aslant  (position  A Bi), 
for  in  the  latter  case  fewer  rays  strike  it.  Largely  for  this 
reason,  the  ground  is  hotter  under  the  noonday  sun  than  it  is 
earlier.  The  higher  temperature  in  summer  than  in  winter  is 
due  to  the  fact  that  the  sun  is,  on  an  average,  more  nearly  over- 
head in  summer  than  in  winter.  The  hottest  part  of  the  day 
is  not  at  noon  as  we  might  at  first  expect,  but  an  hour  or  two 
later.  This  lagging  occurs  because  of  the  time  required  to  warm 
up  the  ground  and  the  air.  A  similar  lagging  occurs  in  the  sea- 
sons, so  that  the  hottest  and  the  coldest  weather  do  not  fall  re- 
spectively on  the  longest  day  (June  21)  and  the  shortest  (Dec. 
22),  but  a  month  or  so  later  as  a  rule. 

The  above-mentioned  sources  are  the  three  main  sources  of 
heat.  There  are  other  minor  sources.  An  electric  current  heats 
a  wire  or  any  other  substance — solid,  liquid,  or  gas — through 


FIG.  109. 

which  it  passes.  This  source  is  of  great  commercial  importance. 
The  condensation  of  water  vapor  produces  a  large  amount  of 
heat,  and  this  heat  is  one  of  the  greatest  factors  in  producing 
wind  storms  as  explained  in  Sec.  223. 

162.  The  Effects  of  Heat. — The  principal  effects  of  heat  are : 
(a)  Rise  in  temperature. 
(6)  Increase  in  size. 

(c)  Change  of  state. 

(d)  Chemical  change. 

(e)  Physiological  effect. 
CD  Electrical  effect. 

(a)  With  but  very  few  exceptions  a  body  becomes  hotter, 
i.e.,  its  temperature  rises,  when  heat  is  applied  to  it.  Excep- 
tions :  If  water  containing  crushed  ice  is  placed  in  a  vessel  on  a 
hot  stove,  the  water  will  not  become  perceptibly  hotter  until 
practically  all  of  the  ice  is  melted.  Further  application  of  heat 


220  MECHANICS  AND  HEAT 

causes  the  water  to  become  hotter  until  the  boil'ng  point  is 
reached,  when  it  will  be  found  that  the  temperature  again  ceases 
to  rise  until  all  of  the  water  boils  away,  whereupon  the  contain- 
ing vessel  becomes  exceedingly  hot.  In  this  case,  the  heat  energy 
supplied,  instead  of  causing  a  temperature  rise  (a),  has  been  used 
in  producing  a  change  of  state  (c),  i.e.,  it  has  been  used  in  changing 
ice  to  water,  or  water  to  steam. 

(6)  As  heat  is  supplied  to  a  body,  it  almost  invariably  produces 
an  increase  in  its  size.  It  might  readily  be  inferred  that  the 
more  violent  molecular  vibrations  which  occur  as  the  body  be- 
comes hotter,  would  cause  it  to  occupy  more  space,  just  as  a 
crowd  takes  more  room  if  the  individuals  are  running  to  and  fro 
than  if  they  are  standing  still  or  moving  less.  Exception  to 
(6) :  If  a  vessel  filled  with  ice  is  heated  until  the  ice  is  melted, 
the  vessel  will  be  only  about  9/10  full.  In  this  case  heat  has 
caused  a  decrease  in  size.  This  case  is  decidedly  exceptional, 
however,  in  that  a  change  of  state  (c)  is  involved.  It  is  also 
true  that  most  substances  expand  upon  melting  instead  of 
contracting  as  ice  does. 

(d)  To  ignite  wood,  coal,  or  any  other  substance,  it  is  neces- 
sary to  heat  it  to  its  "kindling"  or  ignition  temperature,  before 
the  chemical  change  called  "burning"  will  take  place.     In  the 
limekiln,   the   excessive   heat   separates   carbon   dioxide    (CO2) 
from  the  limestone,  or  crude  calcium  carbonate  (CaCO3),  leaving 
calcium  oxide  (CaO),  called  lime.     There  are  other  chemical  re- 
actions besides  oxidation  which  take  place  appreciably  only  at 
high  temperatures.     Slow  oxidation  of  many  substances  occurs 
at  ordinary  temperatures.     All  chemical  reactions  are  much  less 
active  at  extremely  low  temperatures  such  as  the  temperature 
produced  by  liquid  air. 

(e)  Heat  is  essential  to  all  forms  of  life.     Either  insufficient 
heat  or  excessive  heat  is  exceedingly  painful. 

(/)  The  production  of  electrical  effect  by  heat  will  be  discussed 
under  the  head  of  the  Thermocouple  (Sec.  174). 

163.  Temperature. — The  temperature  of  a  body  specifies  its 
state  with  respect  to  its  ability  to  impart  heat  to  other  bodies. 
Thus,  if  a  body  A  is  at  a  higher  temperature  than  another  body 
B,  it  will  always  be  found  that  heat  will  flow  from  A  to  B  if  they 
are  brought  into  contact,  or  even  if  brought  near  together.  The 
greater  the  temperature  difference  between  A  and  B,  other  things 
being  equal,  the  more  rapid  will  be  the  heat  transfer.  The  tern- 


THERMOMETRY  AND  EXPANSION  221 

perature  of  a  body  rises  as  the  heat  vibrations  of  its  molecules 
become  more  violent. 

The  temperature  of  a  body  cannot  be  measured  directly,  but  it 
may  be  measured  by  some  of  the  other  effects  of  heat,  as  (6) 
and  (/)  (Sec.  162),  or  it  may  be  roughly  estimated  by  the  physi- 
ological effect  or  temperature  sense.  Heat  of  itself  always  passes 
from  a  body  of  higher  temperature  to  one  of  lower  tempera- 
ture. The  temperature  sense  serves  usually  as  a  rough  guide 
in  determining  temperature,  but  it  is  sometimes  very  unre- 
liable and  even  misleading,  as  may  be  seen  from  the  following 
examples. 

If  the  right  hand  is  placed  in  hot  water  and  the  left  hand  in 
cold  water  for  a  moment,  and  then  both  are  placed  in  tepid  water, 
this  tepid  water  will  feel  cold  to  the  right  hand  and  warm  to  the 
left  hand.  Under  these  conditions  heat  flows  or  passes  from  the 
right  hand  to  the  tepid  water.  The  tepid  water  being  warmer 
than  the  left  hand,  the  flow  is  in  the  opposite  direction.  Hence 
if  heat  flows  from  the  hand  to  a  body,  we  consider  the  body  to  be 
cold,  while  if  the  reverse  is  true,  we  consider  it  to  be  warm. 
If  A  shakes  B's  hand  and  observes  that  it  feels  cold  we  may  be 
sure  that  B  notices  that  A's  hand  is  warm. 

If  the  hand  is  touched  to  several  articles  which  have  been 
lying  in  a  cool  room  for  some  time,  and  which  are  therefore  at 
the  same  temperature,  it  will  be  found  that  the  articles  made  of 
wool  do  not  feel  noticeably  cool  to  the  touch.  The  cotton  articles, 
however,  feel  perceptibly  cool,  the  wooden  articles  cold,  and  the 
metal  articles  still  colder.  The  metal  feels  colder  than  wood 
or  wool,  because  it  takes  heat  from  the  hand  more  rapidly,  due 
to  its  power  (called  conductivity)  of  transmitting  heat  from  the 
layer  of  molecules  in  contact  with  the  hand  to  those  farther 
away.  Wood  is  a  poor  conductor  of  heat  and  wool  is  a  very 
poor  conductor;  so  that  in  touching  the  latter,  practically  only 
the  particles  .touching  the  hand  are  warmed,  and  hence  very 
little  heat  is  withdrawn  from  the  hand  and  no  sensation  of  cold 
results. 

One  of  the  most  accurate  methods  of  comparing  and  measur- 
ing temperatures,  and  the  one  almost  universally  used,  makes 
use  of  the  fact  that  as  heat  is  supplied  to  a  body,  its  temperature 
rise,  and  its  expansion,  or  increase  in  size,  go  hand  in  hand. 
Thus  if  10°  rise  in  temperature  causes  a  certain  metal  rod  to  be- 
come 1  mm.  longer,  then  an  increase  of  5  mm.  in  length  will 


222  MECHANICS  AND  HEAT 

show  that  the  temperature  rise  is  almost  exactly  5  times  as 
great,  or  practically  50°.  This  principle  is  employed  in  the  use 
of  thermometers. 

164.  Thermometers. — From  the  preceding  section  it  will  be 
seen  that  any  substance  which  expands  uniformly  with  tempera- 
ture rise  can  be  used  for  constructing  a  thermometer.     Air  or 
almost  any  gas,  mercury,  and  the  other  metals  meet  this  require- 
ment and  are  so  used.     Alcohol  is  fairly  good  for  this  purpose 
and  has  the  advantage  of  not  freezing  in  the  far  north  as  mercury 
does.     Water  is  entirely  unsuitable,  because  its  expansion,  as 
its  temperature  rises,  varies  so  greatly.     When  ice  cold  water 
is  slightly  heated  it  actually  decreases  in  volume  (see  Maximum 
Density,  Sec.  185) ;  whereas  further  heating  causes  it  to  expand, 
but  not  uniformly. 

The  fact  that  in  the  case  of  alcohol,  the  expansion  per  degree 
becomes  slightly  greater  as  the  temperature  rises,  makes  it  neces- 
sary to  gradually  increase  the  length  of  the  degree  divisions 
toward  the  top  of  the  scale.  In  the  case  of  mercury,  the  expan- 
sion is  so  nearly  uniform  that  the  degree  divisions  are  made  of 
equal  length  throughout  the  scale. 

Mercury  is  the  most  widely  used  thermometric  substance. 
It  is  well  adapted  to  this  use  because  it  expands  almost  uni- 
formly with  temperature  rise;  has  a  fairly  large  coefficient  of 
expansion;  does  not  stick  to  the  glass;  has  a  low  freezing  point 
(  -38°.8  C.)  and  a  high  boiling  point  (357°  C.) ;  and,  being  opaque, 
a  thin  thread  of  it  is  easily  seen. 

165.  The  Mercury  Thermometer. — The  mercury  thermometer 
consists  of  a  glass  tube  T  (Fig.  110)  of  very  small  bore,  termi- 
nating in  a  bulb  B  filled  with  mercury.     As  the  bulb  is  heated, 
the   mercury  expands   and  rises  in  the  tube  (called  the  stem), 
thereby  indicating  the  temperature  rise  of  the  bulb.     In  filling 
the  bulb,  great  care  must  be  taken  to  exclude  air. 

Briefly,  the  method  of  introducing  the  mercury  into  the  bulb 
is  as  follows:  The  bulb  is  first  heated  to  cause  the  air  contained 
in  it  to  expand,  in  order  that  a  portion  of  it  may  be  driven  out  of 
the  open  upper  end  of  the  stem.  This  end  is  then  quickly  placed 
in  mercury,  so  that  when  the  bulb  cools,  and  consequently  the  air 
pressure  within  it  falls  below  one  atmosphere,  some  mercury  is 
forced  up  into  the  bulb.  If,  now,  the  bulb  is  again  heated  until 
the  mercury  in  it  boils,  the  mercury  vapor  formed  drives  out  all 
of  the  air;  so  that  upon  again  placing  the  end  of  the  stem  in  the 


THERMOMETRY  AND  EXPANSION 


223 


mercury  and  allowing  the  bulb  to  cool,  thereby  condensing  the 
vapor,  the  bulb  and  stem  are  completely  filled  with  mercury. 

Let  us  suppose  that  the  highest  temperature  which  the  above 
thermometer  is  designed  to  read  is  120°  C.  The  bulb  is  heated 
to  about  125°,  expelling  some  of  the  mercury  from 
the  open  end  of  the  tube  which  is  then  sealed  off. 
Upon  cooling,  the  mercury  contracts,  so  that  a  vacuum 
is  formed  in  the  stem  above  the  mercury.  It  will 
be  evident  that  as  the  mercury  in  B  is  heated  and  ex- 
pands, its  upper  surface,  called  its  meniscus  m,  will 
rise;  while  if  it  is  cooled  its  contraction  will  cause 
the  meniscus  to  fall.  Attention  is  called  to  the  fact 
that  if  mercury  and  glass  expanded  equally  upon  be- 
ing heated,  then  no  motion  of  m  would  result.  Mer- 
cury, however,  has  a  much  larger  coefficient  of  expan- 
sion than  glass  (see  table,  Sec.  171).  If  heat  is  sud- 
denly applied,  for  example  by  plunging  the  bulb  into 
hot  water,  the  glass  becomes  heated  first,  and  m 
actually  drops  slightly,  instantly  to  rise  again  as  the 
mercury  becomes  heated. 

The  position  of  the  meniscus  m,  then,  except  in  the 
case  of  very  sudden  changes  in  temperature  such  as 
just  cited,  indicates  the  temperature  to  which  the  bulb 
B  is  subjected.     In  order,  however,  to  tell  definitely 
what  temperature  corresponds  to  a  given  position  of 
m,  it  is  necessary  to  "calibrate"  the  thermometer. 
To  do  this,  the  thermometer  is  placed  in  steam  in  an 
enclosed  space  over  boiling  water.     This  heats  the 
mercury  in  B,  thereby  causing  it  to  expand,  and  the 
meniscus  m  rises  to  a  point  which  may  be  marked 
a.     The  thermometer  is  next  placed  in  moist  crushed 
ice  which  causes  the  mercury  to  contract,  thereby  low- 
ering the  meniscus  to  the  point  marked  6.     We  have        U  B 
now  determined  two  fixed  points,  a  and  b,  corresponding 
respectively  to  the  boiling  point  of  water  and  the    F 
melting  point  of  ice.     It  now  remains  to  decide  what 
we  shall  call  the  temperatures  corresponding  to  a  and  6,  which 
decision  also  determines  how  many  divisions  of  the  scale  there 
shall  be  between  these  two  points.     Several  different  "scales" 
are  used,  two  of  which  will  be  discussed  in  the  next  section. 

Thermometers  should  not  be  calibrated  until  several  years 


224 


MECHANICS  AND  HEAT 


B 


after  filling.  If  calibrated  immediately,  it  will  be  found  after 
a  short  time  that  because  of  the  gradual  contraction  that  has 
taken  place  in  the  glass,  all  of  the  readings  are  slightly  too  high. 
166.  Thermometer  Scales. — The  two  thermometer  scales  in 
common  use  are  the  Centigrade  and 
Fahrenheit  scales.  To  calibrate  a 
thermometer,  according  to  the  cen- 
tigrade scale,  the  point  6  (Fig.  110) 
is  marked  0°,  and  the  point  a  is 
marked  100°,  which  makes  it  nec- 
essary to  divide  ab  into  100  equal 
parts  in  order  that  each  part  shall 
correspond  to  a  degree.  Accord- 
ingly we  see  that  ice  melts  at  zero 
degrees  centigrade,  written  0°  C., 
and  that  water  boils  at  100°  C.  In- 
creasing the  pressure,  slightly  lowers 
the  melting  point  of  ice  (Sec.  186) 
and  appreciably  raises  the  boiling 
point  of  water  (Sec.  194).  To  be 
accurate,  ice  melts  at  0°  C.  and 
water  boils  at  100°  C.  when  sub- 
jected to  standard  atmospheric 
pressure  (76  cm.  of  mercury).  If 
the  pressure  differs  from  this,  cor- 
rection must  be  made,  at  least  in 
the  case  of  the  boiling  point. 

The  Fahrenheit  scale  is  in  com- 
mon use  in  the  United  States  and 
Great  Britain.  To  calibrate  the 
thermometer  (Fig.  110)  according 
to  the  Fahrenheit  scale,  the  "ice 
point"  6  is  marked  32°,  and  the 
boiling  point  a  is  marked  212°. 
The  difference  between  these  two 
points  is  180°  so  that  ab  will  have 
to  be  divided  into  180  equal  spaces 

in  order  that  each  space  shall  correspond  to  a  degree  change  of 
temperature.  Using  the  same  space  for  a  degree,  the  scale  may 
be  extended  above  212  and  below  32. 

The  Fahrenheit  scale  has  the  advantage  of  a  low  zero  point 


FIG.  111. 


THERMOMETRY  AND  EXPANSION  225 

which  makes  it  seldom  necessary  to  use  negative  readings,  and 
small  enough  degree  division  that  it  is  commonly  unnecessary 
to  use  fractional  parts  of  a  degree  in  expressing  temperatures. 
The  Reaumer  scale  ("ice  point"  0°,  "boiling  point"  80°),  used  for 
household  purposes  in  Germany,  has  nothing  to  recommend  it. 

It  is  frequently  necessary  to  change  a  temperature  reading 
from  the  Fahrenheit  scale  to  the  centigrade  or  vice  versa.  For 
convenience  in  illustrating  the  method,  let  A  and  B  (Fig.  Ill) 
represent  two  thermometers  which  are  exactly  alike  except  that 
A  is  calibrated  according  to  the  centigrade  scale,  and  B  accord- 
ing to  the  Fahrenheit.  If  both  are  placed  in  crushed  ice,  A  will 
read  0°  C.  and  B,  32°  F.;  while  if  placed  in  steam,  A  will  read 
100°  C.  and  B,  212°  F.  If  both  thermometers  are  placed  in 
warm  water  in  which  A  reads  40°  C.,  then  the  temperature  / 
that  thermometer  B  should  indicate  may  be  found  as  follows: 
The  fact  that  the  distance  between  the  ice  point  and  boiling  point 
is  100°  on  A,  and  180°  on  B,  shows  that  the  centigrade  degree 
is  180/100  or  9/5  Fahrenheit  degrees.  From  the  figure  it  is 
seen  that  /  is  40°  C.  above  ice  point  or  40X9/5  =  72°  F.  above 
32°  F.,  or  104°  F.  Next,  let  both  thermometers  be  placed  in 
quite  hot  water  in  which  B  reads  140°  F.,  and  let  it  be  required 
to  find  the  corresponding  reading  c  of  A.  Since  140  —  32  =  108, 
the  distance  be  corresponds  to  108°  F.,  or  108X5/9  =  60°  C. 
Hence  140°  F.  =  60°  C.  In  the  same  way  any  temperature 
reading  may  be  changed  from  one  scale  to  the  other. 

167.  Other  Thermometers. — There  are  several  different  kinds  of 
thermometers,  each  designed  for  a  special  purpose,  which  we  shall  now 
briefly  consider. 

Maximum  Thermometer. — In  the  maximum  thermometer  of  Negretti 
and  Zambra  there  is,  near  the  bulb,  a  restriction  in  the  capillary  bore  of 
the  stem.  As  the  temperature  rises,  the  mercury  passes  the  restriction, 
but  as  the  temperature  falls,  and  the  mercury  in  the  bulb  contracts,  the 
mercury  thread  breaks  at  the  restriction  and  thus  records  the  maximum 
temperature.  To  reset  the  instrument,  the  mercury  is  forced  past  the 
restriction  down  into  the  bulb  by  the  centrifugal  force  developed  by 
swinging  the  thermometer  through  an  arc. 

The  Clinical  Thermometer. — The  clinical  thermometer,  used  by  phy- 
sicians, differs  from  the  one  just  described  in  that  it  is  calibrated  for  but 
a  few  degrees  above  and  below  the  normal  temperature  of  the  body 
(98°. 6  F.).  It  also  has  a  large  bulb  in  comparison  with  the  size  of  the 
bore  of  the  stem,  thus  securing  long  degree  divisions  and  enabling  more 
accurate  reading. 


226 


MECHANICS  AND  HEAT 


Six's  Maximum  and  Minimum  Thermometer. — In  this  thermometer 
the  expansion  of  the  alcohol  (or  glycerine)  in  the  glass  bulb  A  (Fig.  112), 
as  the  temperature  rises,  forces  the  mercury  down  in  tube  B  and  up  in 
the  tube  C.  As  the  mercury  rises  in  C  it  pushes  the  small  index  c 

(shown  enlarged  at  left)  before  it. 
When  the  temperature  again  falls,  c 
is  held  in  place  by  a  weak  spring  and 
thus  records  the  maximum  tempera- 
ture. The  contraction  of  the  alco- 
hol in  A  as  the  temperature  decreases 
causes  the  mercury  to  sink  in  C  and 
rise  in  B.  As  the  mercury  rises  in  B 
it  pushes  index  b  before  it  and  thus 
records  the  minimum  temperature. 
This  thermometer  is  convenient  for 
meteorological  observations.  The 
instrument  is  reset  by  drawing  the 
indexes  down  to  the  mercury  by 
means  of  a  magnet  held  against  the 
glass  tube. 

The  Wet-and-dry-bulb  Thermom- 
eter, also  used  in  meteorological 
work,  is  discussed  in  Sec.  198  and 
Sec.  222. 

The  Gas  Thermometer. — There  are 
two  kinds  of  gas  thermometers,  the 
constant-pressure  and  the  constant- 
volume  thermometers.  A  simple 
form  of  Constant-pressure  Thermom- 
eter is  shown  in  Fig.  113.  As  the 
gas  in  B  is  heated  or  cooled,  the 
accompanying  expansion  and  con- 
traction forces  the  liquid  index  /  to 
the  right  or  left.  The  fact  that  for 
each  degree  of  rise  or  fall  in  tem- 
perature, the  volume  of  a  given 
quantity  of  gas  (under  constant  pres- 
sure) changes  by  1/273  of  its  volume 
at  0°  C.  (Sec.  171),  makes  possible 
the  accurate  marking  of  the  degree 

division  on  the  stem,  provided  the  volume  of  B  and  the  cross  section  of 
the  bore  of  the  stem  are  both  known. 

A  simple  form  of  the  Constant-volume  Gas  Thermometer  is  shown  in 
Fig.  114.  The  stem  A  of  the  bulb  B  which  contains  the  gas  is  connected 
with  the  glass  tube  C  by  the  rubber  tube  T  which  contains  the  mercury. 


FIG.  112. 


THERMOMETRY  AND  EXPANSION 


227 


When  a  quantity  of  gas  is  heated  and  not  permitted  to  increase  in 
volume,  its  pressure  increases  1/273  of  its  pressure  at  0°  C.  for  every 
degree  (centigrade  degree)  rise  in  temperature  (Sec.  171).  If,  when  B 
is  at  0°  C.,  and  meniscus  mi  is  at  mark  a,  the  meniscus  m2  is  at  the  same 
level  as  mi,  then  it  is  known  that  the  pressure  of  the  gas  in  B  is  one  atmos- 
phere. If,  now,  the  temperature  of  .B  rises,  mi  is  pushed  down;  but  by 


raising  C  until  m2  is  at  the  proper  height  h  above  mi,  the  mercury  is 
forced  back  to  mark  a,  thus  maintaining  the  constancy  of  the  volume 
of  air  in  B  and  A.  Suppose  that  the  required  height  h  is  10  cm.  The 
excess  pressure  of  the  gas  in  B  above  atmospheric  pressure  will  then  be 
10/76  or  36/273  atmospheres,  and  the  temperature  of  B,  according  to  the 
gas  law  just  stated,  must  be  36°  above  zero,  that  is  36°  C. 

The  Constant-volume  Hydrogen  Ther- 
mometer is  by  international  agreement 
the  standard  instrument  for  tempera- 
ture measurements.  This  instrument 
differs  in  detail,  but  not  in  principle, 
from  the  one  shown  in  Fig.  114. 

The  Dial  Thermometers. — If  the  tube 
of  the  Bourdon  Gage  (Sec.  141)  is  filled 
with  a  liquid  and  then  plugged  at  A, 
the  expansion  of  the  liquid  upon  beir.g 
heated  will  change  the  curvature  of 
the  tube  and  actuate  the  index  just  as 
explained  for  the  case  of  steam  pressure. 

The  Metallic  Thermometer. — A  spiral 
made  of  two  strips  of  metal  a  and 
b  soldered  together  (Fig.  115)  will  un- 
wind slightly  with  temperature  rise  if 
the  metal  b  expands  more  rapidly  than 
a.  As  the  spiral  unwinds  it  causes 
the  index  I  to  move  over  the  scale  and  indicate  the  temperature. 

Recording  Thermometer. — If  the  scale  in  Fig.  115  were  replaced  by  a 
drum  revolving  about  a  vertical  axis  and  covered  by  a  suitably  ruled 
sheet  of  paper,  and  if,  further,  the  left  end  of  the  index  7  were  provided 
with  an  inked  tracing  point  resting  on  the  ruled  sheet,  we  would  then 
have  represented  the  essentials  of  the  recording  thermometer  or  Thermo- 
graph. The  drum  is  driven  by  a  clock  mechanism  and  makes  (usually) 


FIG.  114. 


228  MECHANICS  AND  HEAT 

one  revolution  per  week.  If  the  temperature  remains  constant,  the  trac- 
ing point  draws  a  horizontal  line  on  the  drum  as  it  rotates  under  it.  As 
the  temperature  rises  and  falls,  the  tracing  point  rises  and  falls  and 
traces  on  the  revolving  drum  an  irregular  line  which  gives  a  permanent 
and  continuous  record  of  the  temperature  for  the  week.  Obviously  the 
days  of  the  week,  subdivided  into  hours,  would  be  marked  on  the  sheet 
around  the  circumference  of  the  drum;  while  the  temperature  lines, 
properly  spaced,  would  run  horizontally  around  the  drum  and  be  num- 
bered in  degrees  from  the  bottom  upward. 


FIG.  115. 

168.  Linear  Expansion. — When  a  bar  of  any  substance  is 
heated  it  becomes  slightly  longer.  In  some  cases,  especially 
with  the  metals,  allowance  must  be  made  for  this  change  in 
length,  called  linear  expansion.  Thus,  a  slight  space  is  left  between 
the  ends  of  the  rails  in  railroad  construction.  If  this  were  not 
done,  the  enormous  force  or  end  thrust  exerted  by  the  rails  upon 
expansion  during  a  hot  day  would  warp  the  track  out  of  shape. 
The  contraction  and  expansion  of  the  cables  supporting  large 
suspension  bridges  cause  the  bridge  floor  to  rise  and  fall  a  dis- 
tance of  several  inches  as  the  temperature  changes.  A  long 
iron  girder  bridge  should  have  one  end  free  to  move  slightly 
lengthwise  (on  rollers)  on  the  supporting  pier  to  permit  its  ex- 
pansion and  contraction  without  damage  to  the  pier. 

In  the  familiar  process  of  "shrinking"  hot  iron  tires  onto 
wooden  wagon  wheels,  use  is  made  of  the  contraction  of  the  tire 
that  takes  place  when  it  cools.  Cannons  are  constructed  of 
concentric  tubes,  of  which  the  outer  ones  are  successively  heated 
and  "shrunk"  onto  the  inner  ones.  This  extremely  tight  fitting 


THERMOMETRY  AND  EXPANSION  229 

of  the  outer  layers  insures  that  they  will  sustain  part  of  the  stress 
when  the  gun  is  fired. 

169.  Coefficient  of  Linear  Expansion.  —  When  a  bar,  whose 
length  at  0°  C.  is  L0,  has  its  temperature  raised  to  1°  C.,  its  length 
increases  by  a  certain  fraction  a  of  its  original  length  L0.  This 
fraction  a,  which  is  very  small  for  all  substances,  is  called  the 
coefficient  of  linear  expansion  for  the  material  of  which  the  bar  is 
composed.  The  actual  increase  in  length  of  the  bar  is  then  L0a. 
When  heated  from  0°  to  2°,  the  increase  in  length  is  found  to 
be  almost  exactly  twice  as  great  as  before,  or  L01a;  while  if 
heated  from  0°  to  t°,  it  is  very  closely  L0at.  Consequently  the 
length  of  the  bar  at  any  temperature  t,  which  length  may  be 
represented  by  Lt,  is  given  by  the  equation. 


Lt=Lo+L0at=L0(l  +  a£)  (75) 

whence 


in  which  Lt—L0  is  the  total  increase  in  length  for  a  change  of  t 
degrees,  and  hence  (Lt—  L0)  divided  by  t  is  the  total  change  for 
one  degree.  If  this  total  change  is  divided  by  the  length  L0  of 
the  bar  (in  cms.)  we  have  the  increase  in  length  per  centimeter 
of  length  (measured  at  0°  C.)  per  degree  rise  of  temperature, 
which  by  Eq.  75a  is  a.  Thus  a  may  also  be  defined  as  the  increase 
in  length  per  centimeter  (i.e.,  per  cm.  of  the  length  of  the  bar 
when  at  0°  C.)  produced  by  1°  C.  rise  in  temperature,  or  the 
increase  in  length  per  centimeter  per  degree. 

To  illustrate,  suppose  that  two  scratches  on  a  brass  bar  are 
1  cm.  apart  when  the  bar  is  at  0°  C.  Then,  since  a  for  brass  is 
0.000019  (approx.,  see  table),  it  follows  that  at  1°  C.  the 
scratches  will  be  1.000019  cm.  apart;  at  2°  C.,  1.000038  cm.;  at 
10°  C.,  1.00019  cm.  apart,  etc.  Since  the  length  Lt  of  a  metal 
bar  at  a  temperature  t  differs  very  little  from  its  length  at  0°, 
i.e.,  L0,  we  may  for  most  purposes  consider  that  its  increase  in 
length  when  heated  from  a  temperature  t  to  i+1  is  Lta  instead 
of  L0a.  Consequently,  when  heated  from  a  temperature  t  to 
a  still  higher  temperature  t',  the  increase  in  length  is  approx- 
imately Lt  a  (t'  —  t).  We  then  have  the  length  L/  at  the  higher 
temperature  expressed  approximately  in  terms  of  Lt  by  the 
equation 


230 


MECHANICS  AND  HEAT 


This  equation  is  accurate  enough  for  all  ordinary  work  and  it 
is  also  a  very  convenient  equation  to  use  in  all  problems  involv- 
ing two  temperatures,  neither  of  which  is  zero.  Strictly  speak- 
ing, a  is  not  constant,  but  increases  very  slightly  in  value  with 
temperature  rise. 

AVERAGE  COEFFICIENT  OF  LINEAR  EXPANSION   OF  A  FEW  SUBSTANCES 


Substance 

Coeff.  of  Exp.  a 

Substance 

Coeff.  of  Exp.  a 

Brass 

0  0000185 

Oak,  with  grain. 

0  000005 

Copper  
Glass  
Ice 

0.0000168 
0.0000086 
0  000050 

Platinum  
Quartz,  fused  .... 
Silver 

0.0000088 
0.000005 
0  000019 

Iron  

0.000012 

Zinc  

0.000029 

It  is  perhaps  well  for  the  student  to  memorize  a  for  plati- 
num and  note  that  for  oak  it  is  less  than  for  platinum  and 
for  most  metals  about  twice  as  great.  In  the  case  of  glass,  a 
varies  considerably  for  the  different  kinds. 

The  French  Physicist  Guillaume  recently  made  the  interest- 
ing discovery  that  the  coefficient  of  expansion  of  a  certain  nickel- 
steel  alloy  (36  per  cent,  nickel),  known  as  Invar,  is  only  about 
one-tenth  as  large  as  that  of  platinum,  or  0.0000009.  From  these 
figures  we  see  that  the  length  of  a  bar  of  this  metal  increases 
less  than  1  part  in  1,000,000  when  its  temperature  is  raised 
1°  C.  Steel  tapes  and  standards  of  length  are  quite  commonly 
made  of  Invar. 

170.  Practical  Applications  of  Equalities  and  Differences  in 
Coefficient  of  Linear  Expansion. — In  the  construction  of  incan- 
descent lamps  it  is  necessary  to  have  a  vacuum  in  the  bulb,  or 
the  carbon  filament  that  gives  off  the  light  will  quickly  oxidize 
or  "burn  out."  The  electric  current  must  be  led  through  the 
glass  to  the  filament  by  means  of  wires  sealed  into  the  glass 
while  hot.  If  the  glass  and  wire  do  not  expand  alike  upon  being 
heated,  the  glass  will  crack  and  the  bulb  will  be  ruined.  Plati- 
num wire  is  used  for  this  purpose  because  its  coefficient  of  ex- 
pansion is  almost  exactly  the  same  as  that  of  glass. 

The  differences  between  the  coefficients  of  expansion  for  any 
two  metals,  for  example,  brass  and  iron,  has  many  practical 
applications.  Important  among  the  devices  which  utilize  these 
differences  in  expansion  are  the  automatic  fire  alarm,  the  thermo- 
stat, and  the  mechanism  for  operating  the  "skidoo"  lamp  used  in 
signs.  Another  very  important  application  of  this  difference 


THERMOMETRY  AND  EXPANSION 


231 


in  expansion  of  two  metals  is  in  the  temperature  compensation 
of  clock  pendulums  and  the  balance  wheels  of  watches.  By 
means  of  these  compensation  devices,  timepieces  are  prevented 
from  gaining  or  losing  time  with  change  of  temperature. 

The  Fire  Alarm. — The  operation  of  the  fire  alarm  will  be 
understood  from  a  study  of  Fig.  116.  An  iron  bar  I  and  a  brass 
bar  B  are  riveted  together  at  several  points  and  attached  to  a 
fixed  support  D  at  one  end,  the  other  end  C  being  free.  Since 
the  coefficient  of  expansion  for  brass  is  greater  than  for  iron,  it 
will  be  evident  that  the  above  composite  bar  will  curve  upward 
upon  being  heated,  and  downward  upon  being  cooled.  Conse- 
quently the  end  C  will  rise  when  the  temperature  rises,  and  fall 
when  the  temperature  falls.  If  such  a  device  is  placed  near  the 


Battery 


FIG.  116. 

ceiling  of  a  room,  and  if  by  suitable  wiring,  electrical  connections 
are  made  between  it  and  an  electric  bell,  it  become  a  fire  alarm. 
For  if  a  fire  breaks  out  in  the  room,  both  bars  /  and  B  will  be 
equally  heated,  but  B  will  elongate  more  than  7,  thus  causing 
C  to  rise  until  it  makes  contact  with  P.  This  contact  closes  the 
electrical  circuit  and  causes  the  electric  bell  to  ring. 

The  Thermostat. — If  the  room  above  considered  becomes  too 
cold,  C  descends  and  may  be  caused  to  touch  a  suitably  placed 
point  Pi,  thereby  closing  another  electrical  circuit  (not  shown) 
connected  with  the  mechanism  that  turns  on  more  heat.  As 
soon  as  the  temperature  of  the  room  rises  to  its  normal  value,  C 
again  rises  enough  to  break  connection  with  Plf  and  the  heat 
supply  is  either  cut  off  or  reduced,  depending  upon  the  adjust- 
ment and  design  of  the  apparatus.  When  so  used,  the  above 
bar,  with  its  connections,  is  called  a  thermostat. 


232  MECHANICS  AND  HEAT 

In  a  common  form  of  thermostat,  the  motion  of  C,  when  the 
room  becomes  too  cold,  opens  a  "needle"  valve  to  a  compressed 
air  pipe.  This  pipe  leads  to  the  compressed  air  apparatus, 
which  is  so  arranged  that  when  the  air  escapes  from  the  above- 
mentioned  valve,  more  heat  is  turned  on. 

The  "Skidoo  Lamp." — This  device  is  very  much  used  in 
operating  several  lamps  arranged  so  as  to  spell  out  the  words 
of  a  sign.  Such  a  sign  is  much  more  noticeable  if  the  lamps 
flash  up  for  an  instant  every  few  seconds  than  if  they  shine 
steadily.  The  arrangement  (using  only  one  lamp)  is  shown  in 
Fig.  117.  The  binding  posts  E  and  F  are  connected  to  the  light- 
ing circuit.  Bars  /  and  B  are  arranged  just  as  in  Fig.  116,  ex- 
cept that  the  brass  bar  is  above  the  iron  bar  instead  of  below. 

When  these  bars  are  not  touching  the  point  p,  the  electric 
current  passes  from  E  to  a,  at  which  point  the  wire  is  soldered 


5 


FIG.  117. 

to  the  bars,  then  on  through  the  coil  D  of  very  many  turns  of 
fine  wire  wrapped  about  the  bars,  to  point  P,  where  the  wire  is 
again  soldered,  and  finally  through  the  lamp,  back  to  the  binding 
post  marked  F. 

Since  coil  D  offers  very  great  resistance  to  the  passage  of  cur- 
rent, only  a  small  current  flows,  and  the  lamp  does  not  glow. 
This  small  current,  however,  heats  coil  D  and  therefore  bars  B 
and  /;  and,  since  B  expands  more  rapidly  than  /,  point  C  moves 
down  until  it  touches  point  P  as  explained  in  connection  with 
Fig.  116.  The  instant  that  point  C  touches  P,  practically  all 
of  the  current  flows  directly  from  a  through  the  heavy  bars  to 
P  and  then  through  the  lamp  as  before.  The  fact  that  the 
current  does  not  have  to  flow  through  coil  D  when  C  and  P  are 
in  contact  produces  two  marked  changes  which  are  essential  to 
the  operation  of  the  lamp.  First,  since  the  electrical  resistance 
of  the  bars  is  small,  the  current  is  much  greater  than  before  and 


THERMOMETRY  AND  EXPANSION 


233 


the  lamp  glows;  and  second,  the  coil  now  having  practically 
no  current,  cools  down  slightly,  thus  permitting  the  bars  to  cool 
down,  thereby  causing  C  to  rise.  The  instant  C  rises,  the  current 
is  obliged  to  go  through  the  coil,  and  is  therefore  too  weak  to 
make  the  lamp  glow,  but  it  heats  the  coil,  causing  C  to  descend 
again  and  the  cycle  is  thus  repeated  indefinitely.  If  the  contact 
screw  S  is  screwed  down  closer  to  P,  the  lamp  "winks"  at 
shorter  intervals. 

The  Balance  Wheel  of  a  Watch. — The  same  principle  discussed 
above  is  used  in  the  "temperature  compensation"  of  the  balance 


E 


wheel  of  a  watch,  due  to  which  compensation  its  period  does  not 
change  with  change  of  temperature.  When  an  uncompensated 
wheel  is  heated  the  resulting  expansion  causes  its  rim  to  be 
farther  from  its  axis,  thereby  increasing  its  moment  of  inertia. 
As  its  moment  of  inertia  increases,  the  hairspring  (H.S.,  Fig. 
118)  does  not  make  it  vibrate  so  quickly  and  the  watch  loses 
time.  To  make  matters  worse,  the  hairspring  becomes  weaker 
upon  being  heated. 

It  will  be  noticed  in  the  balance  wheel,  sketched  in  Fig.  118, 
that  the  expansion  produced  by  a  rise  in  temperature  causes  the 
masses  C  and  D  (small  screws)  to  move  from  the  center;  while 


234 


MECHANICS  AND  HEAT 


at  the  same  time  it  causes  E  and  F  to  move  toward  the  center. 
For  the  brass  strip  B  forming  the  outside  of  the  rim  expands 
more  than  the  iron  strip  I  inside.  If  the  watch  runs  faster 
when  warmed  it  shows  that  it  is  overcompensated;  whereas  if 
it  runs  slower  when  warmed  it  is  undercompensated.  Over- 
compensation  would  be  remedied  by  replacing  screws  E  and  F 
by  lighter  ones,  at  the  same  time  perhaps  replac- 
ing C  and  D  by  heavier  ones. 

The  Gridiron  Pendulum. — From  the  sketch  of 
the  gridiron  pendulum  shown  in  Fig.  119,  it  will 
be  seen  that  the  expansion  of  the  steel  strip  a, 
and  the  steel  rods  b,  d,  and/,  causes  the  pendulum 
bob  B  to  lower,  thereby  increasing  the  period  of 
the  pendulum;  whereas  the  expansion  of  the  zinc 
rods  c  and  e  evidently  tends  to  raise  B,  thereby 
shortening  the  pendulum  and  also  its  period 
By  having  the  proper  relation  between  the  lengths 
of  the  zinc  and  the  iron  rods,  these  two  opposing 
tendencies  may  be  made  to  exactly  counterbal- 
ance each  other.  In  this  case  the  period  of  the 
pendulum  is  unaffected  by  temperature  changes, 
that  is,  exact  temperature  compensation  is  ob- 
tained. If  rods  c  and  e  were  brass,  their  upward 
expansion  would  not  compensate  for  the  down- 
ward expansion  of  the  iron  rods.  It  would  then 
be  necessary  to  have  four  rods  of  brass  and  five 
of  iron. 

171.  Cubical  Expansion  and  the  Law  of 
Charles. — When  a  given  quantity  of  any  sub- 
stance, say  a  metal  bar,  whose  volume  at  0°  C.  is 
V0,  has  its  temperature  raised  to  1°  C.,  its  volume  increases  by 
a  certain  small  fraction  0  of  its  original  volume  V0.  This  fraction 
(8  is  called  the  coefficient  of  cubical  expansion  of  the  substance  in 
question.  The  actual  increase  in  volume  is  then  F0/3.  If  the 
bar  is  heated  from  0°  to  t°,  i.e.,  through  t  times  as  great  a  range, 
the  increase  in  volume  is  found  to  be  almost  exactly  t  times  as 
great,  or  V0ftt.  Accordingly,  the  volume  at  t°,  which  may  be 
represented  by  Vt,  is  given  by  the  equation 

Vt  =  V0+ V0&t  =  V0  (1  +00  (76) 

whence  Vt—V0 


FIG.  119. 


V0t 


(77) 


THERMOMETRY  AND  EXPANSION 


235 


In  Eq.  77,  Vt—  V0  is  the  total  increase  in  volume;  (Vt—V0)-*- 
t  is  the  total  increase  per  degree  rise  in  temperature;  and  divid- 
ing the  latter  expression  by  V0  gives  (Vt—  F0)-5- Vj,  or  the  in- 
crease per  degree  per  cubic  centimeter.  But  (Vt—  V0)  +  VJt> 
is  /3  from  Eq.  77.  Hence  /3  is  numerically  the  increase  in  volume 
per  cubic  centimeter  of  the  "original"  volume  per  degree  rise  in 
temperature.  By  "original"  volume  is  meant  the  volume  of  the 
bar  when  at  0°  C. 

Equations  76  and  77  apply  to  volumes  of  solids,  liquids,  or 
gases.  The  values  of  0,  however,  differ  widely  for  different  sub- 
stances, as  shown  in  the  table  below.  These  equations  apply 
to  gases  only  if  free  to  expand  against  a  constant  pressure  when 
heated. 

When  a  solid,  e.g.,  a  metal  bar,  expands  due  to  temperature  rise,  it 
increases  in  each  of  its  three  dimensions — length,  breadth,  and  thick- 
ness. For  this  reason,  it  may  be  shown  that  the  coefficient  of  cubical 
expansion  is  3  times  the  coefficient  of  linear  expansion  for  the  same 
substance;  i.e.,  /3=3«.  For,  consider  a  cube  of  metal,  say,  each  edge 
of  which  has  a  length  L0  at  0°  C.  Then,  by  Eq.  74,  the  length  of 
each  side  at  a  temperature  t°  will  be  L0(l  +  aO.  The  volume  at  0°,  or 
V0,  is  L03;  while  the  volume  V«  at  1°  is 


V>L0 


(78) 


Expanding  (1  +  aO8,  we  have  !+3aZ-(-3a2£2+a3J3.  Now,  since  a  is 
very  small,  a2  and  a3  will  be  negligibly  small  (observe  that  (1/1000)2  = 
1/1,000,000),  and  the  terms  3aH*  and  <*3t3  may  be  dropped.  Eq.  78 
then  becomes 


Vt=V0  (1+3  at) 


(79) 


By  comparing  Eq.  79  with  Eq.  76  we  see  at  once  that  /3=3a.  which 
was  to  be  proved.  In  like  manner  it  may  be  shown  that  the  coefficient 
of  area  expansion  of  a  sheet  of  metal,  for  example,  is  2  a. 

Accordingly,  the  fractional  parts  by  which  the  length  of  a  bar  of  iron, 
the  area  of  a  sheet  of  iron,  and  the  volume  of  a  chunk  of  iron  increase 
per  degree,  are  respectively  0.000012,  0.000024,  and  0.000036. 

COEFFICIENT  OF  CUBICAL  EXPANSION  OF  A  FEW  SUBSTANCES 


Substance 


Substance 


Alcohol  
Ether 

0.00104 
0  0017 

Air,  and  all  gases  
Iron 

0.00367 
0  000036 

Mercury  
Petroleum  

0.00018 
0.00099 

Zinc  
Glass  

0.000087 
0.000026 

236 


MECHANICS  AND  HEAT 


r 


Gas 


The  Law  of  Charles. — If  a  quantity  of  gas  which  is  confined 
in  a.  vessel  A  (Fig.  120)  by  a  frictionless  piston  P,  at  atmospheric 
pressure  and  0°  C.,  is  heated  to  1°  C.  it  will  expand  1/273  (or 
0.00367)  of  its  original  volume;  so  that  its  volume  becomes 
1.00367  times  as  great.  The  fact  that  this  value  of  /3  (Eq.  77) 
is  practically  the  same  for  all  gases  was  discovered  by  Charles 
and  is  known  as  the  Law  of  Charles. 

If,  now,  the  piston  is  prevented  from  moving,  then,  as  the  gas 
is  heated  it  cannot  expand,  but  its  pressure  will  increase  1/273 
for  each  degree  rise  in  temperature,  as  might  be  detected  by  the 
attached  manometer  M;  while  if  cooled  1°,  its  pressure  will  de- 
crease 1/273.  If  cooled  to  10°  below  zero  its  pressure  will  de- 
crease 10/273  of  its  original  value,  etc.  Hence  the  inference, 
that  if  it  were  possible  to  cool  a  gas 
to— 273°  C.  it  would  exert  no  pressure 
whatever. 

Absolute  Zero  and  the  Kinetic  Theory 
of  Gases. — According  to  the  Kinetic 
Theory  of  Gases,  a  gas  exerts  pressure 
because  of  the  to-and-fro  motion  of 
its  molecules  (Sec.  131).  These  mole- 
cules are  continually  colliding  with 
each  other,  and  also  bombarding  the 
walls  of  the  enclosing  vessel.  The  impact  of  the  molecules 
in  this  bombardment  gives  rise  to  the  pressure  of  gases,  just 
as  we  know  that  a  ball,  thrown  against  the  wall  and  then  re- 
bounding from  it,  reacts  by  producing  a  momentary  thrust 
against  the  wall.  Millions  of  such  thrusts  per  second  would, 
however,  give  rise  to  a  steady  pressure.  Under  ordinary  con- 
ditions the  average  speed  of  the  air  molecules  required  to  pro- 
duce a  pressure  of  15  Ibs.  per  sq.  in.  is  about  1400  ft.  per  sec. 
But  a  body  is  supposed  to  have  heat  energy  due  to  the  motion 
of  its  molecules.  It  may  therefore  be  said :  (a)  that  at  —  273°  C. 
a  gas  would  exert  no  pressure  (see  above) ;  hence  (6)  that  its  molec- 
ular motion  must  cease;  and  therefore  (c)  that  it  would  have  no 
heat  energy  at  this  temperature.  When  a  body  has  lost  all  of 
its  heat  energy,  it  cannot  possibly  become  any  colder.  This 
temperature  of  —273°  C.  is  therefore  called  the  Absolute  Zero, 
It  is  interesting  to  note  that  extremely  low  temperatures, 
within  a  few  degrees  of  the  absolute  zero,  have  been  produced 
artificially.  By  permitting  liquid  helium  to  evaporate  in  a  par- 


FIG.  120. 


THERMOMETRY  AND  EXPANSION  237 

tial  vacuum,  Kammerlingh-Onnes  (1908)  produced  a  temperature 
of  —270°  C.,  or  within  3°  of  the  absolute  zero. 

172.  The  Absolute  Temperature  Scale.  —  If  the  above  absolute 
zero  is  taken  as  the  starting  point  for  a  temperature  scale,  then 
on  this  scale,  called  the  Absolute  Centigrade  Scale,  ice  melts  at 
+273°;  water  boils  at  373°  (373°  A.);  a  temperature  of  20°  C.= 
293°  A.,  and  -10°  C.  =  263°  A.,  etc.  This  absolute  scale  is  of 
great  value  from  a  scientific  point  of  view.  Its  use  also  greatly 
simplifies  the  working  of  certain  problems. 

It  will  now  be  shown  that  if  the  pressure  upon  a  gas  is  kept 
constant  while  its  temperature  is  increased  from  T\  to  T2,  then  its 
volume  will  be  increased  in  the  ratio  of  these  two  temperatures 
expressed  on  the  absolute  scale.  In  other  words, 


in  which  Vi  and  F2  represent  the  volume  of  the  gas  at  the  lower 
and  higher  temperatures  respectively,  and  TI  and  Tz,  the  cor- 
responding temperatures  on  the  absolute  scale. 

Proof:  Obviously  77i  =  «i+273,  and  772  =  <2+273;  i.e.,  the 
centigrade  readings  t\  and  tz  are  changed  to  absolute  tempera- 
ture readings  by  adding  273,  which  is  the  difference  between  the 
zeros  of  the  two  scales.  From  Eq.  76,  since  ft  is  1/273,  we  have 


and  likewise  Vz  =  V0(l  +073 

(1+  **  } 
Vz     IV  ^273/ 

=~     " 


V       T 

i.e.,  =?  =  •  z  (pressure  being  kept  constant)  (80) 

V\     i  i 

Eq.  80  shows  that  if  the  absolute  temperature  of  a  certain 
quantity  of  gas  is  made  say  5/4  as  great,  its  volume  becomes 
5/4  as  great;  while  if  the  absolute  temperature  is  doubled  the 
volume  is  doubled,  etc.  It  must  be  borne  in  mind  that  Eq.  80 
holds  only  in  case  the  gas,  when  heated,  is  free  to  expand  against 
a  constant  pressure.  A  discussion  of  Fig.  121  will  make  clear 


238  MECHANICS  AND  HEAT 

the  application  of  Eq.  80.  Let  A  be  a  quantity  of  gas  of  volume 
Fi  and  temperature  27°  C.  confined  in  a  cylinder  by  a  frictionless 
piston  of  negligible  weight.  Let  the  upper  surface  of  the  piston 
be  exposed  to  atmospheric  pressure.  The  gas  in  A  will  then  also 
be  under  atmospheric  pressure  regardless  of  temperature  change. 
For,  as  the  gas  is  heated,  it  will  expand  and  push  the  piston  up- 
ward; the  pressure,  however,  will  be  unchanged  thereby,  i.e., 
the  pressure  will  be  constant,  and  therefore  Eq.  80  will  apply. 
Next  let  the  gas  in  A  be  heated  from  27°  C.  to  127°  C.,  i.e.,  from 
300°  A.  to  400°  A.  Since  the  absolute  temperature  is  4/3  as 
great  as  before,  we  see  from  Eq.  80  that 


_t o— - *__ 


21°  C. 


P,  will  be  raised  to  a  position  PI  such 
that  the  volume  of  the  gas  will  be  4/3  of 
its  former  volume.  Experiment  will 
show  that  the  new  volume  is  4/3  times 
the  old,  thus  verifying  the  equation.  Let 
J  us  again  emphasize  the  fact  that  the  two 

?  volumes  are  to  each  other  as  the  corre- 
sponding absolute  temperatures,  not  centi- 
grade temperatures. 

Since,  as  above  stated,  the  pressure  of 

I          a  body  of  gas  that  is  not  permitted  to 


JPIG    121  expand  increases  i/273  of  its  value  when 

the  gas  is  heated  from  0°  to  t°  C.,  it  fol- 
lows that  the  pressures  p\  and  p2  corresponding  to  the  tempera- 
tures ti  and  £2,  are  given  in  terms  of  p0  (the  pressure  when  the 
temperature  is  zero)  by  the  equations 

pl  =  p0(l+2j^  and  ?*"^1"hj7a) 
from  which  (see  derivation  of  Eq.  80)  we  have 

rn 

=  TTT  I  volume  constant)  (81) 

Pi     L  i 

This  equation  shows  that  if  any  body  of  gas,  contained  in  a  rigid 
vessel  to  keep  its  volume  constant,  has  its  absolute  temperature 
increased  in  a  certain  ratio,  then  its  pressure  will  be  increased  in 
the  same  ratio.  • . 

Boyle's  Law  is  expressed  in  Eq.  72  as  pV  =  K.  Consequently 
if  the  pressure  on  the  gas  in  question  is  increased  to  p\  the  volume 
will  decrease  to  Fi,  but  the  product  will  still  be  K;  i.e.,  p\V\  =  K. 
Likewise  pzVi  =  K,  and  therefore  p\V\  =  pzVz  or  Vz/V\  = 


THERMOMETRY  AND  EXPANSION  239 

Summarizing,  we  may  write  the  three  important  gas  laws, 
namely  Boyle's,  Charles's,  and  the  one  referring  to  pressure 
variation  with  temperature,  thus: 

(E-£),  <72bis> 


Observe  that  the  subscript  T  indicates  that  Eq.  72  is  true  only 
if  the  gas  whose  pressure  and  volume  are  varied  is  maintained  at 
a  constant  temperature.  The  subscript  p  of  Eq.  80  indicates  that 
the  pressure  to  which  the  gas  is  subjected  must  not  vary,  and  V 
of  Eq.  81,  that  the  volume  must  not  vary. 

Attention  is  called  to  the  fact  that  the  three  important  variables 
of  the  gas,  namely  pressure,  volume,  and  temperature,  might 
all  change  simultaneously.  If  the  temperature  of  the  gas  is  kept 
constant,  Boyle's  Law  (Eq.  72)  states  that  the  volume  varies 
inversely  as  the  applied  pressure.  Eq.  80  states  that  if  the  pressure 
upon  the  gas  is  kept  constant,  then  the  volume  varies  directly 
as  the  absolute  temperature;  while  Eq.  81  states  that  if  the  volume 
of  the  gas  is  kept  constant,  then  the  pressure  varies  directly  as 
the  absolute  temperature. 

The  General  Case. — In  case  both  the  temperature  of  a  gas  and 
the  pressure  to  which  it  is  subjected  change,  then  the  new  volume 
(note  that  all  three  variables  change)  may  easily  be  found  by 
considering  the  effect  of  each  change  separately;  i.e.,  by  suc- 
cessively applying  Boyle's  Law  and  Charles's  Law.  To  illus- 
trate, let  the  volume  of  gas  in  A  (Fig.  121),  when  at  atmospheric 
pressure  and  20°  C.,  be  400  cm.3,  and  let  it  be  required  to  find  its 
volume  if  the  pressure  is  increased  to  2|  atmospheres,  and  its 
temperature  is  raised  to  110°  C.  The  new  pressure  is  5/2 
times  the  old;  hence,  due  to  pressure  alone,  in  accordance  with 
Boyle's  Law,  the  volume  will  be  reduced  to  2/5X400  cm.3  The 
original  temperature  of  20°  C.  is  293°  A.,  and  the  new  tempera- 
ture is  383°  A.;  hence,  due  to  the  temperature  change  alone,  the 
volume  would  be  383/293X400  cm.3  Considering  both  effects, 
the  new  volume  would  then  be 

2     ^8*3 

7  =  ^X00^X400  cm.3 
o     ^yo 


240 


MECHANICS  AND  HEAT 


We  may  proceed  in  a  similar  manner  if  both  the  volume  and  the 
temperature  are  changed,  and  the  new  pressure  that  the  gas  will 
be  under  is  required  in  terms  of  the  old  pressure. 

173.  The  General  Law  of  Gases. — We  shall  now  develop  the 
equation  expressing  the  relation  between  the  old  and  the  new 
values  of  pressure,  volume,  and  temperature  of  some  confined 
gas  when  all  three  of  these  quantities  are  changed.  Let  1,  2, 
and  3,  respectively,  be  the  initial,  second,  and  final  positions 
of  the  piston  A  (Fig.  122).  In  the  initial  state,  A  confines  a 
certain  quantity  of  gas  of  volume  V0, 
pressure  p0  (say  1  atmosphere),  and  tem- 
perature To  (say  0°  C.  or  273°  A). 

The  second  state  is  produced  by  heat- 
ing the  gas  from  T0  to  T,  in  which  T/T0 
'      expressed  in  the  absolute  scale  is,  say, 
r     about  3/2.     This  change  in  temperature 
causes  the  gas  to  expand   against   the 
constant  pressure  p0  until  A  is  at  2,  the 
new  volume  V  being  about  f   V0.     In 
this  second  state  of  the  gas,  its  condition 
V,  and  T  as  indicated  in  the  sketch,  and, 


FIG.  122. 


is  represented  by  p0 
from  Eq.  80,  we  have 


_        _ 

Vo~T0'  ~2V 

The  third  state  of  the  gas,  represented  by  p,  V,  and  T,  is  pro- 
duced by  placing  a  weight  on  A,  thereby  increasing  the  pres- 
sure from  p0  to  p  (as  sketched  p/p0  =  5/4  approx.),  and  push- 
ing the  piston  from  position  2  to  its  final  position  at  3,  and 
consequently  reducing  the  volume  from  V  to  V  (as  sketched 
V/V  =  4/5  approx.).  From  Boyle's  Law  (Eq.  72  bis,  just  given), 


Substituting    in  this    equation    the  value  of    V  given  above, 
we  have 


-TT  -TT   ' 

poV'  =  ~°T7T=~^[ 


that  is, 


(82) 


THERMOMETRY  AND  EXPANSION  241 

PoV0 

in  which  R  is  equal  to  ^-o°'  an^  *s  therefore  a  known  constant 

Zi  o 

if  p  and  V0  are  known.  Obviously,  if  twice  as  great  a  mass 
of  the  same  gas,  or  an  equal  mass  of  some  other  gas  half 
as  dense,  were  placed  under  the  piston,  the  constant  R  would  then 
become  twice  as  large. 

Eq.  82  expresses  the  General  Law  of  Gases,  and  is  called  the 
General  Gas  Equation.  From  this  general  equation,  we  see  (a) 
that  for  a  given  mass  of  gas  the  volume  varies  inversely  as  the  pres- 
sure if  the  temperature  is  constant  (Boyle's  Law)  ;  (6)  that  the 
volume  varies  directly  as  the  absolute  temperature  T  if  the  pres- 
sure p  is  constant  (Law  of  Charles);  and  (c)  that  the  pressure 
varies  directly  as  the  absolute  temperature  if  the  volume  V  is 
constant.  The  law  embodied  in  (c)  has  not  received  any  name. 

Let  us  now  use  Eq.  82  to  work  the  problem  given  under  the 
heading  "The  General  Case"  (Sec.  172).  Let  us  represent  the 
first  state  by  ptVi  =  RTi  and  the  second  state  by  p2V2 


. 

Pi  p* 

Consequently 

V2     RT2  .  RTi 

Vi~   p2    '    pi 
or 

7^X^,0^x11x400 
Pz      i  i  5     293 

as  before  found.  Let  us  again  emphasize  the  fact  that  T,  T\, 
and  T2  represent  temperatures  on  the  absolute  scale. 

174.  The  Thermocouple  and  the  Thermopile.  —  If  a  piece  of 
iron  wire  /  (Fig.  123)  has  a  piece  of  copper  wire  C  fastened  to 
each  end  of  it  as  shown,  it  will  be  found  that  if  one  point  of  con- 
tact of  these  two  dissimilar  metals,  say,  B  is  kept  hotter  than  the 
other  junction  A,  an  electric  current  will  flow  in  the  direction 
indicated  by  the  arrows.  This  current  might  be  measured  by 
the  instrument  D.  If  B  is,  say,  60°  hotter  than  A,  the  electric 
current  will  be  about  6  times  as  large  as  if  it  is  only  10°  hotter. 
Two  such  junctions  so  used  constitute  a  Thermocouple.  Any  two 
different  metals  may  be  used  for  a  thermocouple.  Antimony 
and  bismuth  give  the  strongest  electrical  effect  for  a  given  dif- 
ference of  temperature  between  junctions. 

One  hundred  or  so  thermocouples,  made  of  heavy  bars  and 


242  MECHANICS  AND  HEAT 

properly  connected,  form  a  Thermobattery  of  considerable  strength. 
The  greatest  usefulness  of  thermocouples,  however,  is  in  delicate 
temperature  measurements  by  means  of  the  thermopile. 

The  Thermopile. — By  observing 
the  readings  of  D  while  the  tem- 
perature difference  between  A  and 
B  is  varied  through  a  considerable 
range  (Fig.  123),  in  other  words, 
by  calibrating  the  thermocouple,  it 
becomes  a  thermometer  for  meas- 
uring temperature  differences.  A 
large  number  of  such  thermocouples 
properly  connected  constitute  a 
FIG.  123.  Thermopile,  which  will  detect  ex- 

ceedingly small  differences  of  tem- 
perature. The  thermopile  readily  detects  the  heat  radiated  from 
the  hand,  or  from  a  lighted  match,  at  a  distance  of  several  feet. 

PROBLEMS 

1.  Express  60°  C.  and  -30°  C.  on  the  Fahrenheit  scale,  and  also  on  the 
absolute  scale. 

2.  Express  200°  A.  on  the  Fahrenheit  scale,  and  also  on  the  centigrade 
scale. 

3.  An  iron  rail  is  32  ft.  long  at  0°  C.     How  long  is  it  on  a  hot  day  when  at 
40°  C.? 

4.  A  certain  metal  bar,  which  is  3  meters  in  length  at  20°  C.,  is  0.30  cm. 
longer  at  100°  C.     Find  a  for  this  metal. 

6.  If  the  combined  lengths  of  the  iron  rods  a,  b,  and  d  (Fig.  119)  is  100  cm., 
how  long  must  c  and  e  each  be  to  secure  exact  temperature  compensation? 

6.  How  many  H.P.  does  the  sun  expend  upon  one  acre  at  noon?     Assume 
the  sun  to  be  directly  overhead. 

7.  The  cavity  of  a  hollow  brass  sphere  has  a  volume  of  800  cm.3  at  20°  C. 
What  is  the  volume  of  the  cavity  at  50°  C.? 

8.  If  600  cm.3  of  gas,  at  20°  C.  and  atmospheric  pressure,  is  heated  to 
40°  C.,  and  is  free  to  expand  by  pushing  out  a  piston  against  the  pressure  of 
the  atmosphere,  what  will  be  its  new  volume? 

9.  If  6  cu.  ft.  of  air,  at  20°  C.  and  atmospheric  pressure,  is  compressed 
until  its  volume  is  2  cu.  ft.,  and  is  then  heated  to  300°  C.,  what  will  be  its 
new  pressure? 


CHAPTER  XIV 
HEAT  MEASUREMENT,  OR  CALORIMETRY 

175.  Heat  Units. — Before  taking  up  the  discussion  of  the 
measurement  of  quantity  of  heat,  it  will  be  necessary  to  define 
the  unit  in  which  to  express  quantity  of  heat.  The  unit  most 
commonly  used  is  the  Calorie.  The  calorie  may  be  roughly 
defined  as  the  quantity  of  heat  required  to  raise  the  temperature 
of  one  gram  of  water  1°  C.  To  be  accurate,  the  actual  tempera- 
ture of  the  water  should  be  stated  in  this  definition,  since  the 
quantity  of  heat  required  varies  with  the  temperature.  Thus, 
the  quantity  of  heat  required  to  raise  the  temperature  of  1  gram 
of  water  through  a  range  of  1°  is  greater  at  0°  than  at  any  other 
temperature,  and  almost  1  per  cent,  greater  than  it  is  at  20°,  at 
which  point  it  is  a  minimum. 

Some  authors  select  this  range  from  0°  C.  to  1°  C.,  others 
3°.5  C.  to  4°.5  C.,  4°  C.  to  5°  C.,  etc.,  which  gives  of  course 
slightly  different  values  for  the  calorie.  In  selecting  15°  C.  to 
16°  C.  as  the  range,  we  have  a  calorie  of  such  magnitude  that  100 
calories  are  required  to  raise  the  temperature  of  one  gram  of 
water  from  0°  C.  to  100°  C.  Hence  the  calorie  is  perhaps  best 
defined  as  the  amount  of  heat  required  to  raise  the  temperature  of 
one  gram  of  water  from  15°  C.  to  16°  C. 

In  the  British  system,  unit  quantity  of  heat  is  the  quantity 
required  to  raise  the  temperature  of  1  Ib.  of  water  1°  F.,  and  is 
called  the  British  Thermal  Unit,  or  B.T.U.  Since  heat  is  a  form 
of  energy,  it  may  be  expressed  in  energy  or  work  units.  One 
B.T.U.  =778  ft.-lbs.  This  means  that  778  ft.-lbs.  of  work 
properly  applied  to  1  Ib.  of  water,  for  example,  in  stirring  the  water, 
will  raise  its  temperature  1°  F.  From  the  above  statement, 
since  1  Ib.  of  water  in  falling  778  ft.  develops  778  ft.-lbs.  of 
energy,  we  see  that  if  a  1-lb.  mass  of  water  strikes  the  ground  after 
a  778-ft.  fall,  and  if  it  were  possible  to  have  all  of  the  heat  developed 
by  the  impact  used  in  heating  the  water,  then  this  heat  would 
raise  its  temperature  1°  F.  In  fact  this  temperature  rise  is  in- 
dependent of  the  quantity  of  water,  and  depends  solely  upon  the 
height  of  fall.  For,  while  the  heat  energy,  developed  by,  say  3 

243 


244  MECHANICS  AND  HEAT 

Ibs.  of  water,  due  to  impact  after  a  778-ft.  fall,  would  be  3  times  as 
much  as  above  given,  the  amount  of  water  to  be  heated  would 
also  be  3  times  as  much,  and  the  resulting  temperature  rise 
would  therefore  be  1°  F.  as  before.  The  calorie  is  4.187  X107 
ergs.  That  is,  if  4.187  XlO7  ergs  of  energy  are  used  in  stirring 
one  gram  of  water,  its  temperature  will  rise  1°  C.  This  4. 187  X 
107  ergs  is  often  called  the  Mechanical  Equivalent  of  heat.  The 
mechanical  equivalent  in  the  English  system  is  778  ft.-lbs. 

176.  Thermal  Capacity. — The  thermal  capacity  of  a  body  is 
denned  as  the  number  of  calories  of  heat  required  to  raise  the 
temperature  of  the  body  1°  C.,  or  it  is  the  amount  of  heat  the 
body  gives  off  in  cooling  1°  C.     It  is  clear  that  a  large  mass  would 
have  a  greater  thermal  capacity  than  a  small  mass  of  the  same 
substance.     That  mass  is  not  the  only  factor  involved  is  shown  by 
the  following  experiment. 

If  a  kilogram  of  lead  shot  at  100°  C.  is  mixed  with  a  kilogram  of 
water  at  0°  C.,  the  temperature  of  the  mixture  will  not  be  50°, 
but  about  3°.  The  heat  given  up  by  the  kilogram  of  lead  in 
cooling  97°  barely  suffices  to  warm  the  1  kilogram  of  water  3°. 
In  fact  the  thermal  capacity  of  the  water  is  about  33  times  as 
great  as  that  of  the  lead;  consequently,  if  33  kilos  of  lead  had  been 
used  in  the  experiment  the  temperature  of  the  mixture  would  have 
been  50°.  The  very  suggestive  and  convenient  term  "water 
equivalent"  is  sometimes  used  instead  of  thermal  capacity. 
Multiplying  the  mass  of  a  calorimeter  by  its  specific  heat  gives 
its  thermal  capacity  or  the  number  of  calories  required  to  warm  it 
one  degree.  Suppose  that  this  number  is  60.  Now  60  calories 
would  also  heat  60  grams  of  water  one  degree;  hence  the  "water 
equivalent"  of  the  calorimeter  is  60;  i.e.,  the  calorimeter  requires 
just  as  much  heat  to  raise  its  temperature  a  given  amount  as 
would  60  gm.  of  water  if  heated  through  the  same  range. 

177.  Specific  Heat. — -The  Specific  Heat  (s)  of  a  substance  may 
be  defined  as  the  number  of  calories  required  to  heat  1  gm.  of  the 
substance  1°  C.     It  is  therefore  the  thermal  capacity  per  gram  of 
the  substance.     This,  we  see  from  the  definition  of  the  calorie,  is 
practically  equal  to  the  ratio  of  the  heat  required  to  heat  a  given 
mass  of  the  substance  through  a  given  ran^e  of  temperature,  to 
the  heat  required  to  heat  an  equal  mass  of  water  through  the 
same  range.     Thus,  the  specific  heat  of  lead  is  0.031.     This  means 
that  it  would  require  0.031  calorie  to  heat  a  gram  of  lead  one 
degree;  which  is  only  0,031  times  as  much  heat  as  would  be  re- 


HEAT  MEASUREMENT,  OR  CALORIMETRY     245 


quired  to  heat  a  gram  of  water  one  degree.  The  specific  heat  of  a 
substance  is  sometimes  defined  as  the  ratio  just  given.  Since  the 
specific  heat  (calorie  per  gram  per  degree)  of  water  varies  with  the 
temperature  (Sec.  175),  this  definition  lacks  definiteness  as 
compared  with  the  one  we  are  here  using. 

The  table  below  gives  the  specific  heat  of  a  few  substances. 
From  the  values  given,  we  see  that  one  calorie  of  heat  imparted 
to  a  gram  of  glass  would  raise  its  temperature  5°  C.,  while  the 
same  amount  of  heat  imparted  to  a  gram  of  lead  would  raise  its 
temperature  1/0.031,  or  about  32°.5.  In  popular  language  it 
might  be  said  that  lead  heats  6.5  times  as  easily  as  glass,  and  32.5 
times  as  easily  as  water. 

The  specific  heat  of  a  substance  is  usually  expressed  in  calories 
per  gram  per  degree.  Thus,  the  specific  heat  of  lead  is  0.031  cal. 
per  gm.  per  deg.  It  may  also  be  written  0.031  B.T.U.'s  per  Ib. 
per  degree,  the  degree  in  this  case,  however,  being  the  Fahrenheit 
degree.  The  proof  that  the  numeric  (0.031)  is  the  same  in  both 
cases  may  be  left  as  an  exercise  for  the  student. 

The  specific  heat  of  most  substances  varies  considerably  with 
the  temperature.  In  some  cases  there  is  a  decrease  in  its  value 
with  temperature  rise,  while  in  others  there  is  an  increase.  In 
the  case  of  water  the  specific  heat  decreases  up  to  20°  C.  and  then 
increases.  The  values  given  in  the  table  for  the  different  sub- 
stances are  average  values,  taken  at  ordinary  temperatures  (ex- 
cepting in  the  case  of  ice  and  steam). 


Substance 

Sp.  heat  in 
cal.  per 
gm.  per  deg. 

Substance 

Sp.  heat  in 
cal.  per 
gm.  per  deg. 

Brass  
Copper  
Glass 

0.088 
0.093 
0  200 

Ice  
Steam  
Water 

0.504 
0.4  approx. 
1  000  (15°  to  16°) 

Lead  .  
Iron  

0.031 
0.11 

Alcohol  
Petroleum  

0.60 
0.51 

To  heat  a  gram  of  any  substance  of  specific  heat  s  sufficiently 
to  cause  a  temperature  rise  of  t  degrees  requires  st  calories, 
i.e.,  t  times  as  much  heat  as  to  cause  a  rise  of  1  degree.  Further, 
to  heat  M  grams  t  degrees  requires  M  times  as  much  heat  as  to  heat 
one  gram  t  degrees,  or  Mst  calories;  hence,  the  general  expression 
for  the  heat  H  required  to  heat  a  body  of  mass  M  and  specific 
heat  s  from  a  temperature  t\  to  a  temperature  tz,  is 

td  (83) 


246  MECHANICS  AND  HEAT 

If  the  substance  cools  through  this  same  range,  then  H  is  the  heat 
given  off. 

178.  The  Two  Specific  Heats  of  a  Gas.— In  general,  a  body 
when  heated,  expands,  and  in  expanding  it  does  work  in  pushing 
back  the  atmosphere.     This  work  makes  it  require  additional 
heat  energy  to  warm  the  body,  and  therefore  makes  the  specific 
heat  of  the  body  larger  than  it  would  have  been  had  expansion  not 
occurred.      In  case  a  compressed  gas  is  permitted  to  expand 
into  a  space  at  lower  pressure,  the  above  heat  energy  is  taken  from 
the  gas  itself  and  chills  it  greatly.     This  fact  is  utilized  in  the 
manufacture  of  liquid  air  (Sec.  205). 

In  the  case  of  solids  and  liquids,  this  expansion  upon  being  heated 
is  inappreciable,  but  with  gases  it  is  very  great.  Consequently 
the  specifice  heat  of  a  gas,  i.e.,  the  number  of  calories  required  to 
heat  one  gram  one  degree,  is  less  if  the  gas  is  confined  in  a  rigid 
vessel  than  if  it  is  allowed  to  expand  against  constant  pressure 
when  heated.  The  latter  is  called  the  specific  heat  at  constant 
pressure,  and  is  0.237  for  air;  while  the  former  is  called  the  specific 
heat  at  constant  volume,  and  is  0.168  for  air.  The  ratio  of  the  two 
specific  heats  of  air  is  0.237/0.168,  or  1.41.  This  ratio  differs  for 
the  various  gases. 

179.  Law  of  Dulong  and  Petit. — Dulong  and  Petit,  in  1819,  found  by 
experiment  that  for  thirty  of  the  elements,  the  product  of  the  atomic 
weight  and  the  specific  heat  (in  the  solid  state)  is  approximately  constant. 
This  so-called  constant  varies  from  about  6  to  6.6.     For  a  considerable 
number  of  the  elements  it  is  6.4.     For  gases  this  constant  is  about  3.4. 
This  law  does  not  hold  for  liquids,  and  there  are  a  few  solids  that  do  not 
follow  it  at  all  closely. 

Let  us  now  utilize  this  law  in  finding  the  specific  heat  of  iron  and  gold, 
whose  atomic  weights  are  respectively  56  and  196.  The  mathematical 
statement  of  the  law  of  Dulong  and  Petit  is: 

Sp.  heat  X  atomic  weight  =6.4  (approximately)          (84) 

Whence  the  specific  heat  of  gold  is  6.4/196  or  0.0326,  and  that  of  iron 
6.4/56  or  0  114.  These  computed  values  of  the  specific  heat  are  almost 
exactly  the  same  as  those  found  experimentally  for  iron  and  gold. 

The  above  law  shows  that  it  takes  the  same  amount  of  heat  to  warm  an 
atom  one  degree  whether  it  be  a  gold  atom,  an  iron  atom,  or  an  atom  of 
any  other  substance  which  follows  this  law.  For,  from  Eq.  84,  it  is 
obvious  that  if  the  atomic  weight  of  one  element  is  three  times  a£  great 
as  that  of  another  (compare  gold  with  iron),  then  its  specific  heat  must 
be  1/3  as  great  in  order  to  give  the  same  product — 6.4.  But  if  the 


HEAT  MEASUREMENT,  OR  CALORIMETRY     247 

atomic  weight  is  three  times  as  great  for  the  first  metal  as  for  the  second, 
then  the  number  of  atoms  per  gram  will  be  1/3  as  great,  which  accounts 
for  the  first  having  1/3  as  great  specific  heat  as  the  second,  provided  we 
assume  the  same  thermal  capacity  for  all  atoms. 

180.  Specific  Heat,  Method  of  Mixtures.— A  method  which  is 
very  commonly  used  for  determining  the  specific  heat  of  sub- 
stances is  that  known  as  the  method  of  mixtures.  The  method 
can  be  best  explained  in  connection  with  the  apparatus  used,  one 
form  of  which  is  shown  in  section  in  Fig.  124.  H  is  a  heater  con- 
taining some  water  and  having  a  tube  T  passing  obliquely  through 
it  as  shown.  This  tube  contains  the  substance,  e.g.,  the  shot,  the 
specific  heat  of  which  is  to  be  determined.  D  is  a  calorimeter, 
usually  of  brass,  containing  some  water  E. 


FIG.  124. 


First,  the  shot,  the  calorimeter  D,  and  the  water  E,  are  weighed. 
Let  these  masses  be  M ,  M i,  and  M 2,  respectively.  Next  the  water 
in  H  is  heated  to  the  boiling  point  and  kept  boiling  for  a  few 
minutes.  The  steam  surrounding  T  soon  warms  it  and  the 
contained  shot  to  100°  C.,  which  may  be  determined  by  thermom- 
eter C,  thrust  through  cork  B.  The  cork  A  is  now  withdrawn, 
and  the  hot  shot  is  permitted  to  fall  into  the  water  E  to  which  it 
rapidly  imparts  its  heat  until  D,  E,  and  the  shot  are  all  at  the 
same  temperature.  Let  this  temperature  be  t',  and  let  the  tem- 
perature of  E  before  the  shot  was  introduced  be  t.  The  heat 
Hi,  which  the  shot  loses  in  cooling  from  100°  to  t',  is  evidently 


248  MECHANICS  AND  HEAT 

equal  to  the  heat  H%  which  the  calorimeter  and  water  gain  in 
rising  in  temperature  from  t  to  t1  ',  that  is 

#1  =  HZ  (85) 

provided  no  heat  passes  from  the  calorimeter  to  the  air  or  vice 
versa  during  the  mixing  process. 

This  interchange  of  heat  between  the  calorimeter  and  the  air 
cannot  be  totally  prevented,  but  the  error  arising  from  this  cause 
is  largely  eliminated  by  having  D  and  E  a  few  degrees  lower  than 
the  room  temperature  at  the  beginning  of  the  mixing  process  and 
a  few  degrees  higher  than  room  temperature  at  the  end;  i.e., 
after  D,  E,  and  the  shot  have  come  to  the  same  temperature. 
During  the  mixing  process,  the  contents  of  the  calorimeter  should 
be  stirred  to  insure  a  uniform  temperature  throughout. 

Almost  always  in  calorimetric  work,  it  is  assumed  that  the 
heat  given  up  by  the  hot  body  is  equal  to  the  heat  taken  up  by 
the  cold  body;  so  that  Eq.  85  is  the  starting  point  for  the  deriva- 
tion of  the  required  equation  in  all  such  cases.  It  is  much  better 
to  learn  how  to  apply  this  general  equation  than  to  try  to  mem- 
orize special  forms  of  it.  One  such  application  will  be  made  here. 

If  s,  si,  and  s2  represent  the  specific  heats  of  the  shot,  calorim- 
eter, and  water  repectively,  then  from  Eq.  83,  the  heat  given 
up  by  the  shot  is  Jlfs(100—  t')  ;  that  taken  up  by  the  calorimeter  is 
MiS^t'  —  t);  and  that  taken  up  by  the  water  is  MzSz(tr  —  t). 
Since  s2  is  unity  it  may  be  omitted,  and  we  have  from  Eq.  85 

(t'-t)  (85a) 


The  quantities  M,  MI,  and  M%  are  determined  by  weighing,  and 
the  three  temperatures  are  read  from  thermometers,  so  that  the 
one  remaining  unknown,  s,  may  be  solved  for. 

181.  Heat  of  Combustion.  —  Chemical  changes  are,  in  general, 
accompanied  by  the  evolution  of  heat;  a  few,  however,  absorb 
heat.  Most  chemical  salts  when  dissolved  in  water  cool  it,  in 
some  cases  quite  markedly.  In  still  other  cases  solution  is  at- 
tended by  the  development  of  heat.  A  complete  study  of  these 
subjects  is  beyond  the  scope  of  this  volume,  but  the  particular 
chemical  change  known  as  combustion  is  so  all-important  in 
connection  with  commercial  heating  and  power  development 
that  a  brief  discussion  of  it  will  be  given. 

Combustion  is  usually  denned  as  the  violent  chemical  combina- 
tion of  a  substance  with  oxygen  or  chlorine,  and  is  accompanied 


HEAT  MEASUREMENT,  OR  CALORIMETRY     249 


by  heat  and  light.  In  a  more  restricted  sense  it  is  what  is  popu- 
larly known  as  "burning"  which  practically  amounts  to  the 
chemical  combination  of  oxygen  with  hydrogen  or  carbon. 

In  scientific  work,  the  Heat  of  Combustion  of  any  substance  is 
the  number  of  calories  of  heat  developed  by  the  complete  com- 
bustion of  1  gram  of  that  substance.  In  engineering  practice 
it  is  the  number  of  B.T.U.'s  developed  by  the  complete  combus- 
tion of  1  pound  of  a  substance.  The  latter  gives  9/5  as  large  a 
number  as  the  former  for  the  same  substance.  Hence  it  is  neces- 
sary in  consulting  tables  to  determine  whether  the  metric,  or 
the  British  system  is  used.  Obviously,  the  burning  of  one 
gram  of  coal  would  heat  just  as  many  grams  of  water  1°  C.  as 
the  burning  of  a  pound  of  coal  would  heat  pounds  of  water  1°  C. 
But  to  heat  a  pound  of  water  1°  C.  takes  9/5  B.T.U.'s,  since 
1°  C.  equals  9/5°  F.  In  the  following  table,  in  which  the  approxi- 
mate values  of  the  heat  of  combustion  are  given  in  both  sys- 
tems, it  will  be  observed  that  the  numerical  values  are  in  the  above 
ratio  of  9  to  5. 

HEAT  OF  COMBUSTION  WITH  OXYGEN 


Substance 

Product 

Calories  per 
gram 

B.T.U.'s  per 
pound 

Hydrogen(H)  

H2O  

34,000 

61,000 

Carbon  (C)  

CO2  

7,800 

14,000 

Marsh  gas  (CH4) 

CO2  and  H2O 

13  100 

23  600 

Alcohol  (ethyl)  
Petroleum  
Soft  coal  | 

CO2andH2O.. 
CO2andH2O.. 

Mainly    CO2, 

7,200 
11,000 
7,500  to  8,500 
7  800 

13,000 
20,000 
Ave.  14,500 
Ave  14  000 

Wood                              j 

H2O  and  ash. 

4,000  to  4  500 

Ave     7  600 

Dynamite 

1,300 

Iron                  .              ... 

Fe3O4  

1,600 

Zinc  

ZnO  

1,300 

Copper  

CuO  

600 

Hydrogen,  it  will  be  seen,  produces  far  more  heat  per  gram 
than  any  other  substance,  indeed  over  four  times  as  much  as  its 
nearest  rival,  carbon.  Coal  averages  about  the  same  as  carbon. 
Petroleum  contains  hydrogen  combined  with  carbon  (hydrocar- 
bons) and  gives,  therefore,  a  higher  heat  of  combustion  than  pure 
carbon  does.  The  main  gases  that  are  produced  in  the  com- 
bustion of  all  substances  known  as  fuels  are  water  vapor  (H2O) 
and  carbon  dioxide  (CO2). 

It  would  be  well  to  memorize  the  values  in  the  last  column  for 


250  MECHANICS  AND  HEAT 

petroleum,  coal,  and  wood.  Observe  that  dynamite  has  a  sur- 
prisingly low  heat  of  combustion.  Its  effectiveness  as  an  ex- 
plosive depends  upon  the  suddenness  of  combustion  due  to  the 
fact  that  the  oxygen  is  in  the  dynamite  itself,  and  not  taken  from 
the  air  as  in  ordinary  combustion. 

To  find  how  much  chemical  potential  energy  in  foot-pounds 
exists  in  1  Ib.  of  coal,  multiply  14,500  by  778;  i.e.,  multiply  the 
number  of  B.T.U.'s  per  pound  by  the  number  of  foot-pounds 
in  one  B.T.U.  To  reduce  this  result  to  H.P.-hours,  divide  by 
550X3600  (1  hr.  equals  3600  sec.).  Due  to  various  losses  of 
energy  in  the  furnace,  boiler,  and  engine  (Chap.  XVIII),  a  steam 
engine  utilizes  only  about  5  or  10  per  cent,  of  this  energy,  so  that 
the  H.P.-hours  above  found  should  be  multiplied  by  0.05  or  0.10 
(depending  upon  the  efficiency  of  the  engine  used)  to  obtain  the 
useful  work  that  may  be  derived  from  a  pound  of  coal.  With 
a  very  good  furnace,  boiler,  and  engine,  about  1.5  Ibs.  of  coal 
will  do  1  H.P.-hr.  of  work.  Thus  it  would  require  about  150 
Ibs.  of  coal  to  run  a  100-H.P.  engine  for  an  hour. 

182.  Heat  of  Fusion  and  Heat  of  Vaporization. — As  stated  in 
Sec.  162,  considerable  heat  may  be  applied  to  a  vessel  contain- 
ing ice  water  and  crushed  ice  without  producing  perceptible 
temperature  rise  until  the  ice  is  melted,  whereupon  further 
application  of  heat  causes  the  water  to  become  hotter  and  hotter 
until  the  boiling  point  is  reached,  when  the  temperature  again 
ceases  to  rise.  Other  substances  behave  in  much  the  same  way 
as  water.  These  facts  show  that  heat  energy  is  required  to 
change  the  substance  from  the  solid  to  the  liquid  state,  and  from 
the  liquid  to  the  vapor  state.  This  heat  energy  is  supposed  to 
be  used  partly  in  doing  internal  work  against  molecular  forces. 
In  case  the  change  of  state  is  accompanied  by  an  increase  in 
volume,  part  of  this  heat  energy  is  used  in  doing  external  work  in 
causing  the  substance  to  expand  against  the  atmospheric  pressure. 

The  Heat  of  Fusion  of  a  substance  is  the  number  of  calories 
required  to  change  a  gram  of  that  substance  from  the  solid  to 
the  liquid  state  without  causing  a  rise  in  temperature.  The 
Heat  of  Vaporization  is  the  number  of  calories  required  to  change 
a  gram  of  the  substance  from  the  liquid  to  the  vapor  state  at  a 
definite  temperature  and  at  atmospheric  pressure.  These  two 
changes  absorb  heat  while  the  reverse  changes,  that  is  from  vapor 
to  liquid  and  from  liquid  to  solid,  evolve  heat.  The  amounts  of 
heat  evolved  in  these  reverse  changes  are  the  same  respectively 


HEAT  MEASUREMENT,  OR  CALORIMETRY     251 

as  the  amounts  absorbed  in  the  former  changes.  This  equality 
should  be  expected,  of  course,  from  the  conservation  of  energy. 

For  water,  the  heat  of  fusion  is  79.25  calories  per  gram  (also 
written  79.25  cal./gm.),  and  the  heat  of  vaporization  is  536.5 
cal.  pergm.;  which  means  that  to  change  one  gram  of  ice  at  0° 
C.  to  water  at  0°  C.  requires  79.25  calories,  and  to  change  1  gm. 
of  water  at  100°  to  steam  at  100°  and  atmospheric  pressure 
requires  536.5  cal.  The  value  of  the  latter  depends  very  much 
upon  the  temperature.  To  change  a  gram  of  water  at  20°  to 
vapor  at  20°  requires  585  cal.,  in  other  words,  the  heat  of  vapori- 
zation of  water  at  20°  C.  is  585  cal.  per  gm. 

From  reasoning  analogous  to  that  used  in  changing  the  heat  of 
combustion  from  the  metric  to  the  British  system  (Sec.  181),  we 
see  that  the  above  heat  of  vaporization  multiplied  by  9/5  gives 
the  heat  of  vaporization  in  the  British  system,  namely,  966 
B.T.U.'s  per  pound.  That  is  to  say,  966  B.T.U.'s  are  required 
to  change  1  Ib.  of  water  at  212°  F.  to  steam  at  the  same  tempera- 
ture. The  heat  of  fusion  is  rarely  expressed  in  the  British 
system. 


HEAT  OF  FUSION  OF  VARIOUS  SUBSTANCES 


Substance 

Melting 
tem- 
perature 

Calories 
per  gram 

Substance 

Melting 
tem- 
perature 

Calories 
per  gram 

Ice 

0°C. 

79  25 

Silver  .  .   . 

960°  C. 

21 

Ice  
Nitrate  of  soda.  .  . 
Paraffin  

-6 
306 
52 

76 
65 
35 

Cadmium  
Sulphur  
Lead  

315 
115 
325 

13.7 
9.37 
5.86 

Zinc  

415 

28 

Mercury  

-38.8 

2.82 

HEAT  OF  VAPORIZATION  OF  VARIOUS  SUBSTANCES 


Substance 

Tem- 
perature 

Calories 
per  gram 

Substance 

Tem- 
perature 

Calories 
per  gram 

Water  . 

0°C. 
100 
17 

78 
448 

595 
536.5 
295 
206 
362 

Ether  
Mercury  

35°  C. 
357 

61 
0 
30.8 

90 
62 
58.5 
56 
3  72 

Water  
Ammonia  (NH3).. 
Alcohol  (ethyl)  .  .  . 
Sulphur  

Chloroform  
Carbon  dioxide. 
Carbon  dioxide. 

183.  Bunsen's  Ice  Calorimeter. — A  very  sensitive  form  of  ice 
calorimeter  is  that  of  Bunsen,  in  which  the  amount  of  ice  melted  is 
determined  from  the  accompanying  change  of  volume.  It  con- 


252 


MECHANICS  AND  HEAT 


sists  of  a  bulb  A  (Fig.  125),  with  a  tube  B  attached,  and  a  test 
tube  C  sealed  in  as  shown.  The  space  between  A  and  C  is 
completely  filled  with  water  except  the  lower  portion,  which 
contains  mercury  as  does  also  a  portion  of  B. 

By  pouring  some  ether  into  C  and  then  evaporating  it  by  forcing 
a  stream  of  air  through  it  (Sec.  197),  some  ice  E  is  formed  about 
C.  As  this  ice  forms,  expansion  occurs,  which  forces  the  mercury 
farther  along  in  B  to,  say,  point  a.  Next,  removing  all  traces  of 
ether  from  C,  drop  in  a  known  mass  of  hot  substance  D  at  a 
known  temperature  i' .  The  heat  from  D  melts  a  portion  of  the 
ice  E,  and  the  resulting  contraction 
causes  the  mercury  to  recede,  say  to 
a'.  The  volume  of  the  tube  between 
a  and  a'  is  evidently  the  difference  be- 
tween the  volume  of  the  ice  melted  by 
D  and  that  of  the  resulting  water 
formed;  and  hence,  if  known,  could  be 
used  to  determine  the  amount  of  ice 
melted.  Multiplying  this  amount  by 
79.25  would  give  the  number  of  cal- 
ories of  heat  given  off  by  D  in  cooling 
to  0°  C. 

A  simpler  method,  however,  is  to 
calibrate  the  instrument  by  noting  the 
distance,  say  aa" ,  that  the  mercury  col- 
umn recedes  when  one  gram  of  water  at 
100°  is  introduced  into  C.  Suppose 

this  is  two  inches.  Then,  since  the  gram  of  water  in  cooling  to 
0°  C.  would  impart  to  the  ice  100  calories,  we  see  that  a  motion 
of  one  inch  corresponds  to  50  calories.  Accordingly,  the  distance 
aa'  in  inches,  multiplied  by  50,  gives  the  number  of  calories 
given  off  by  D  in  cooling  from  t'  to  zero.  This  enables  the  cal- 
culation of  the  specific  heat  of  the  substance  D. 

184.  The  Steam  Calorimeter. — Dr.  Joly  invented  a  very  sen- 
sitive calorimeter,  known  as  the  Joly  Steam  Calorimeter,  in  which 
the  amount  of  heat  imparted  to  a  given  specimen  in  raising  its 
temperature  through  a  known  range,  is  determined  from  the 
amount  of  steam  that  condenses  upon  it  in  heating  it.  A  speci- 
men whose  specific  heat  is  sought,  e.g.,  a  piece  of  ore  A  (Fig.  126), 
is  suspended  in  an  inclosure  B  by  a  wire  W  passing  freely  through 
a  small  hole  above,  and  attached  to  one  end  of  the  beam  of  a  sensi- 


FIG.  125. 


HEAT  MEASUREMENT,  OR  CALORIMETRY     253 

tive  beam  balance.  Weights  are  added  to  the  other  end  of  the 
beam  until  a  "balance"  is  secured.  As  steam  is  admitted  to  the 
inclosure,  it  condenses  upon  the  ore  until  the  temperature  of  the 
ore  is  100°,  whereupon  condensation  ceases.  The  additional 
weight  required  to  restore  equilibrium,  multiplied  by  536.5,  gives 
the  number  of  calories  required  to  heat  the  ore  and  pan  from  a 
temperature  t  (previously  noted)  to  100°.  For  it  is  evident  that 
each  gram  of  steam  that  condenses  upon  the  ore  imparts  to  it 
536.5  calories.  If  the  mass  of  the  ore  is  known,  its  specific  heat 
can  readily  be  computed  (Eqs.  83  and  85). 

The  pan  in  which  the  ore  is  placed  catches  the  drip,  if  any. 
Obviously  the  amount  of  steam  that  would  condense  upon  the 
pan  in  the  absence  of  the  ore  must  be  found,  either  by  calculation 
or  by  experiment,  and  be  subtracted  from  the  total.  By  the  use 


FIG.  126. 

of  certain  refinements  and  modifications  which  will  not  be 
discussed  here,  the  instrument  may  be  employed  for  very  delicate 
work,  such  as  the  determination  of  the  specific  heat  of  a  com- 
pressed gas  contained  in  a  small  metal  sphere. 

185.  Importance  of  the  Peculiar  Heat  Properties  of  Water. — 
The  fact  that  the  specific  heat,  heat  of  fusion,  and  heat  of 
vaporization  of  water  are  all  relatively  large  is  of  the  utmost 
importance  in  influencing  the  climate.  It  is  also  of  great 
importance  commercially.  From  the  conservation  of  energy  it 
follows  that  if  it  takes  a  large  amount  of  heat  (heat  absorbed)  to 
warm  water,  to  vaporize  it,  or  to  melt  ice;  then  an  equally  large 
amount  of  heat  will  be  given  off  (evolved)  when  these  respective 
changes  take  place  in  the  reverse  sense — that  is,  when  water 
cools,  vapor  condenses,  or  water  freezes, 


254  MECHANICS  AND  HEAT 

Specific  Heat. — In  connection  with  the  subject  of  specific  heat, 
it  is  seen  that  the  amount  of  heat  a  given  mass  absorbs  in  being 
warmed  through  a  given  range  of  temperature  depends  upon  its 
specific  heat.  From  this  fact  it  is  evident  that  a  body  of  water 
would  change  its  temperature  quickly  with  change  of  tempera- 
ture of  the  air,  if  its  specific  heat  were  small.  The  specific 
heat  of  water  is  much  larger  than  for  most  other  substances,  as 
may  be  seen  from  the  table  (Sec.  177).  Note  also  that  water 
has  about  twice  as  large  a  specific  heat  as  either  ice  or  steam. 
Because  of  the  large  specific  heat  of  water  it  warms  slowly  and 
cools  slowly;  so  that  during  the  heat  of  the  day  a  lake  cools  the 
air  that  passes  over  it,  while  in  the  cool  of  the  night,  it  warms  the 
air.  This  same  effect  causes  the  temperature  on  islands  in  mid- 
ocean  to  be  much  less  subject  to  sudden  or  large  changes  than 
it  is  in  inland  countries. 

Heat  of  Fusion. — It  requires  79.25  calories  to  melt  1  gram  of 
ice;  hence,  according  to  the  conservation  of  energy,  a  gram  of 
water  must  give  off  approximately  80  calories  of  heat  when  it 
changes  to  ice.  If  the  heat  of  fusion  were  very  small,  say  2 
calories  per  gram,  a  river  would  not  need  to  give  off  nearly  so 
much  heat  in  order  to  change  to  ice,  so  that  it  might,  under  those 
conditions,  freeze  solid  in  a  night  with  disastrous  consequences 
to  the  fish  in  it,  and  to  the  people  dependent  upon  it  for  water 
supply.  Under  these  circumstances,  it  would  also  be  necessary 
to  buy  about  40  tunes  as  much  ice  to  get  the  same  cooling  effect 
that  we  now  obtain. 

Heat  of  Vaporization. — Since  it  requires  about  600  calories 
to  change  a  gram  of  water  at  ordinary  temperatures  to  vapor, 
it  follows,  from  the  conservation  of  energy,  that  when  a  gram  of 
vapor  condenses  to  water  it  gives  off  about  600  calories  of  heat. 
This  heat,  freed  by  the  condensation  of  vapor,  is  one  of  the  main 
causes  of  winds.  The  heat  developed  causes  the  air  to  become 
lighter,  whereupon  it  rises,  and  the  surrounding  air  as  it  rushes 
in  is  called  a  wind  (Sec.  223). 

If  the  heat  of  vaporization  of  water  were  much  smaller, 
evaporation  and  cloud  formation  would  be  much  more  rapid, 
resulting  ultimately  in  dried  rivers  and  ponds,  alternating 
with  disastrous  floods. 

The  increase  in  volume  which  accompanies  the  freezing  of  water 
is  of  the  utmost  importance  in  nature.  If  ice  were  more  dense 
than  water,  it  would  sink  to  the  bottom  when  formed,  and  our 


HEAT  MEASUREMENT,  OR  CALORIMETRY     255 

shallow  ponds  and  our  rivers  would  readily  freeze  solid.  As  it  is, 
the  ice,  being  less  dense,  remains  at  the  surface,  and  thus  forms  a 
sheath  that  protects  the  water  and  prevents  rapid  cooling. 

The  Maximum  Density  of  water  occurs  at  4°  C.  If  water  at 
this  temperature  is  either  heated  or  cooled  it  expands,  and  con- 
sequently becomes  less  dense.  Hence  in  winter,  as  the  surface 
water  of  our  lakes  becomes  cooler  and  therefore  denser,  it  settles 
to  the  bottom,  and  other  water  that  takes  its  place  is  likewise 
cooled  and  settles,  thus  establishing  convection  currents  (Sec. 
208).  Through  this  action  the  temperature  of  the  entire  lake 
tends  to  become  4°  C.  At  least  it  cannot  become  colder  than 
this  temperature,  for  as  soon  as  any  surface  water  becomes  colder 
than  4°  it  becomes  less  dense,  and  therefore  remains  on  the  sur- 
face and  finally  freezes.  As  soon  as  the  convection  currents 
cease,  the  chilling  action  practically  ceases,  so  far  as  the  deeper 
strata  of  water  are  concerned,  for  water  is  a  very  poor  conductor 
of  heat. 

186.  Fusion  and  Melting  Point. — The  Fusion  of  a  substance 
is  the  act  of  melting  or  changing  from  the  solid  to  the  liquid  state, 
and  the  Melting  Point  is  the  temperature  at  which  fusion  occurs. 
The  melting  point  of  ice  is  a  perfectly  definite  and  sharply  defined 
temperature;  for  which  reason  it  is  universally  used  as  one  of  the 
standard  temperatures  in  thermometry.  Amorphous  or  non- 
crystalline  substances,  such  as  glass  and  resin,  upon  being  heated, 
change  to  a  soft  solid  or  to  a  viscous  liquid,  and  finally,  when 
considerably  hotter,  become  perfectly  liquid.  Such  substances 
have  no  well-defined  melting  point. 

Solutions  of  solids  in  liquids  have  a  lower  freezing  point  than  the 
pure  solvent,  and  the  amount  of  lowering  of  the  freezing  point  is, 
as  a  rule,  closely  proportional  to  the  strength  of  the  solution. 
It  might  also  be  added  that  the  dissolved  substance  also  raises 
the  boiling  point.  For  example,  a  24  per  cent,  brine  freezes  at 
—22°  C.  and  boils  at  about  107°.  Many  other  substances  dis- 
solved in  water  produce  the  same  effect,  differing  in  degree  only. 
Solvents  other  than  water  are  affected  in  the  same  way. 

Alloys,  which  may  be  looked  upon  as  a  solution  of  one  metal  in 
another,  behave  like  solutions  with  regard  to  lowering  of  the  melt- 
ing point.  Thus  Rose's  metal,  consisting  by  weight  of  bismuth 
4  parts,  lead  1,  and  tin  1,  melts  at  94°  C.  and  consequently  melts 
readily  in  boiling  water.  Wood's  metal — bismuth  4,  lead  2, 
tin  1,  and  cadmium  1 — melts  at  70°.  Solder,  consisting  of  lead 


256 


MECHANICS  AND  HEAT 


37  per  cent.,  and  tin  63  per  cent.,  melts  at  180°  C.  Using  either 
a  greater  or  smaller  percentage  of  lead  raises  the  melting  point 
of  the  solder.  In  all  these  cases,  the  melting  point  of  the  alloy 
is  far  lower  than  that  of  any  of  its  components,  as  may  be  seen 
by  consulting  the  accompanying  table. 

TABLE  OF  MELTING  POINTS 


Substance 


Temperature 


Substance  Temperature 


Hydrogen  
Nitrogen 

-255°  C. 
-210 

Lead  
Zinc 

325°  C. 
415 

Mercury  
Ice 

-  38.8 
0 

Salt  (NaCl)  
Silver 

800 
960 

Phosphorus  
Tin  
Bismuth  
Cadmium.  .  . 

44 
233 
267 
315 

Gold  
Iron  
Platinum  
Iridium.  .  . 

1064 
1200  to  1600 
1755 
2300 

Supercooling. — It  is  possible  to  cool  water  and  other  liquids 
several  degrees  below  the  normal  freezing  point  before  freezing 
occurs.  Thus  water  has  been  cooled  ten  or  twenty  degrees  below 
zero,  but  the  instant  a  tiny  crystal  of  ice  is  dropped  into  the  water, 
freezing  takes  place,  and  the  heat  evolved  (79.25  cal.  per  gm.  of 
ice  formed)  rapidly  brings  its  temperature  up  to  zero.  Dufour 
has  shown  that  small  globules  of  water,  immersed  in  oil,  may 
remain  liquid  from  —20°  C.  to  178°  C.  Some  other  substances, 
e.g.,  acetamid  and  "hypo"  (sodium  hyposulphite),  are  not  so 
difficult  to  supercool  as  is  water. 

Pressure. — Some  substances  when  subjected  to  great  pressure 
have  their  melting  point  raised,  while  others  have  it  lowered. 
Clearly,  if  a  substance  in  melting  contracts  (e.g.,  ice,  Sec.  187),  we 
would  expect  pressure  to  aid  the  melting  process,  and  hence 
cause  the  substance  to  melt  at  a  lower  temperature  than  normal. 
It  has  been  determined,  both  by  theory  and  by  experiment,  that 
ice  melts  at  0.0075°  C.  lower  temperature  for  each  additional 
atmosphere  of  pressure  exerted  upon  it.  This  effect  is  further 
discussed  under  Regelation  (Sec.  188)  and  Glaciers  (Sec.  189). 

187.  Volume  Change  During  Fusion. — Some  substances 
expand  during  fusion,  while  others  contract.  Thus,  in  changing 
from  the  liquid  to  the  solid  state,  water  expands  9  per  cent.,  and 
bismuth  2.3  per  cent.;  while  the  following  contract,  silver 
(10  per  cent.),  zinc  (10  per  cent.),  cast  iron  (1  per  cent.).  Obvi- 
ously silver  and  zinc  do  not  make  good,  clear-cut  castings  for  the 
reason  that  in  solidifying  they  shrink  away  fiom  the  mold.  Silver 


HEAT  MEASUREMENT,  OR  CALORIMETRY     257 


and  gold  coins  have  the  impressions  stamped  upon  them.  Iron 
casts  well  because  it  shrinks  but  slightly.  The  importance  in 
nature  of  the  expansion  of  water  upon  freezing  has  already  been 
discussed  (Sec.  185). 

188.  Regelation.— If  a  block  of  ice  B  (Fig.  127)  has  resting 
across  it  a  small  steel  wire  w,  to  each  end  of  which  is  attached  a 
heavy  weight,  it  will  be  found  that  the  wire  slowly  melts  its  way 
through  the  ice.     The  ice  immediately  below  the  wire  is  subjected 
to  a  very  high  pressure  and  therefore  melts  even  if  slightly  below 
zero  (Sec.  186).     The  water  thus  formed  is  very  slightly  below 
0°  C.,  and  flows  around  above  the  wire  where  it  again  freezes,  due 
to  the  fact  that  it  is  now  at  ordinary  pressure,  and  that  the  sur- 
rounding ice  is  also  a  trifle  below  0°  C.     Thus 

the  wire  passes  through  the  ice  and  leaves  the 
block  as  solid  as  ever.  The  refreezing  of  the 
water  as  it  passes  from  the  region  of  high  pres- 
sure is  called  Regelation.  Since  every  gram  of 
ice  melted  below  the  wire  requires  about  80  cal- 
ories of  heat,  and  since  this  heat  must  come 
from  the  surrounding  ice,  we  see  why  the  ice 
above  the  wire  and  the  water  and  ice  below 
are  cooled  slightly  below  0°  C. 

If  two  irregular  pieces  of  ice  are  pressed  to- 
gether, the  surface  of  contact  will  be  very  small 
and  the  pressure   correspondingly  great;  as  a 
result  of  which  some  of  the  ice  at  this  point  will 
melt.     The  water  thus  formed,  being  at  ordinary  atmospheric 
pressure  and  slightly  below  zero  as  just  shown,  refreezes  and 
firmly  unites  the  two  pieces  of  ice.     A  similar  phenomenon  occurs 
in  the  forming  of  snow  balls  by  the  pressure  of  the  hand. 

In  skating,  regelation  probably  plays  an  important  role,  as 
pointed  out  by  Dr.  Joly.  With  a  sharp  skate,  the  skater's  weight 
bears  upon  a  very  small  surface  of  ice,  which  may  cause  it  to  melt 
even  though  several  degrees  below  zero.  Thus  the  skate  melts 
rather  than  wears  a  slight  groove  in  the  ice.  If  the  ice  is  very 
cold  the  skate  will  not  "bite,"  i.e.,  it  will  not  melt  a  groove, 
unless  very  sharp.  Friction  is  also  probably  much  reduced  by  the 
film  of  water  between  the  skate  and  the  ice. 

189.  Glaciers. — Glaciers    are    great   rivers   of  ice  that  flow 
slowly  down  the  mountain  gorges,  sometimes  (in  the  far  north) 
reaching  the  sea,  where  they  break  off  in  huge  pieces  called  ice- 


FIG.  127 


258  MECHANICS  AND  HEAT 

bergs,  which  float  away  to  menace  ocean  travel.  Glaciers  owe 
both  their  origin  and  their  motion,  in  part,  to  regelation.  Due 
to  the  great  pressure 'developed  by  the  accumulated  masses  of 
snow  in  the  mountains  or  in  the  polar  regions,  part  of  the  snow  is 
melted  and  frozen  together  as  solid  ice,  forming  glaciers,  just  as 
the  two  pieces  of  ice  mentioned  above  were  frozen  together. 

As  the  glacier  flows  past  a  rocky  cliff  that  projects  into  it,  the 
ice  above,  although  at  a  temperature  far  below  zero,  melts  because 
of  the  high  pressure,  flows  around  the  obstacle,  and  freezes  again 
below  it.  The  velocity  of  glaciers  varies  from  a  few  inches  a  day 
to  ten  feet  a  day  (Muir  Glacier,  Alaska),  depending  upon  their 
size  and  the  slope  of  their  beds.  The  mid-portion  of  a  glacier 
flows  faster  than  the  edge  and  the  top  faster  than  the  bottom, 
evidencing  a  sort  of  tar-like  viscosity. 

Glaciers  in  the  remote  past  have  repeatedly  swept  over  vast 
regions  of  the  globe,  profoundly  modifying  the  soil  and  topogra- 
phy of  those  regions.  The  northern  half  of  the  United  States 
shows  abundant  evidence  of  these  ice  invasions  (see  Geology). 
At  present,  glaciers  exist  only  in  high  altitudes  or  high  latitudes. 

190.  The  Ice  Cream  Freezer. — Experiments  show  that  ice, 
in  the  presence  of  common  salt,  may  melt  at  a  temperature  far 
below  0°  C.  (-22°  C.  or  -7°.4  F.).  This  fact  makes  possible 
the  production  of  very  low  temperatures  by  artificial  means. 
The  most  familiar  example  of  the  practical  application  of  this 
principle  is  the  ice  cream  freezer.  The  broken  ice,  mixed  with 
salt,  surrounds  an  inner  vessel  which  contains  the  cream.  The 
rotation  of  the  inner  vessel  serves  the  two-fold  purpose  of  agitat- 
ing the  cream  within,  and  mixing  the  salt  and  ice  without.  The 
revolving  vanes  within  aerate  the  cream,  thus  making  it  light  and 
"velvety."  The  freezing  would  take  place,  however,  without 
revolving  either  vanes  or  container,  but  the  process  would  require 
more  time,  and  the  product  would  be  inferior.  As  the  ice  melts, 
the  water  thus  formed  dissolves  more  salt,  and  the  resulting  brine 
melts  more  ice,  and  so  on.  One  part  (by  weight)  of  salt  to 
three  parts  of  crushed  ice  or  snow  gives  the  best  results.  This 
is  the  proper  proportion  to  form  a  saturated  brine  at  that  low 
temperature. 

The  theory  of  the  production  of  low  temperatures  by  freezing 
mixtures,  such  as  salt  and  ice,  is  very  simple.  Every  gram  of  ice 
that  melts  requires  79.25  calories  of  heat  to  melt  it.  If  this  heat 
is  supplied,  by  a  flame  for  example,  the  temperature  remains  at 


HEAT  MEASUREMENT,  OR  CALORIMETRY     259 

0°  C.  until  practically  all  of  the  ice  is  melted.  If  the  melting  of 
the  ice  is  caused  by  the  presence  of  some  salt  or  other  chemical, 
the  requisite  79.25  calories  of  heat  for  each  gram  melted  must  come 
from  the  freezing  mixture  itself,  and  from  its  surroundings,  mainly 
the  inner  vessel  of  the  freezer,  thus  causing  a  fall  of  temperature. 
Still  lower  temperatures  may  be  obtained  with  a  mixture  of  cal- 
cium chloride  and  snow.  The  cheapness  of  common  salt,  and  the 
fact  that  —22°  C.  is  sufficiently  cold  for  rapid  freezing,  accounts 
for  its  universal  use.  In  fact,  while  being  frozen,  that  is,  while 
being  agitated,  the  cream  should  be  but  a  few  degrees  below 
zero  to  secure  the  maximum  "lightness." 

>  PROBLEMS 

1.  How  much  heat  would  be  required  to  change ^.Q^m.  of  ice  at.  —10°  C 
to  water  at  20°  C.? 

2.  Now  much  heat  wcyuld  be^equired  to  change  40  gm.  of  water  at  30°  C. 
to  steam  at  140°  C.?     Ae  heat  of  vaporization  at  140°  C.  is  about  510  cal. 
per  gm. 

3.  If  40  gm.  of  water  at  80°  C.  is  mixed  with  30  gm.  of  water  at  20°  C., 
what  will  be  the  temperature  of  the  mixture?     Neglect  the  heat  capacity  of 
the  calorimeter.     Suggestion:    Call  the  required  temperature  t,  and  then 
solve  for  it. 

4.  Find  the  "water  equivalent"  of  a  brass  calorimeter  that  weighs  150 
gm. 

6.  Same  as  problem  3,  except  that  the  heat  capacity  of  the  calorimeter 
containing  the  cold  water  is  considered.  The  weight  of  the  calorimeter  is 
60  gm.,  and  the  specific  heat  of  the  material  of  which  it  is  composed  is  0.11. 

6.  A  certain  calorimeter,  whose  water  equivalent  is  20,  contains  80  gm.  of 
water  at  40°  C.     When  a  mass  of  200  gm.  of  a  certain  metal  at  100°  C.  is 
introduced,  the  temperature  of  the  water  and  the  calorimeter  rises  to  55°  C. 
Find  the  specific  heat  of  the  metal. 

7.  How  many  B.T.U.'s  would  be  required  to  change  100  Ibs.  of  ice  at  12°  F. 
to  water  at  80°  F.?     (Sees.  181  and  182.) 

8.  How  many  B.T.U.'s  would  be  required  to  change  100  Ibs.  of  water  at 
80°  F.  to  steam  at  320°  F.?     When  the  water  in  the  boiler  is  heated  to  320°  F. 
the  steam  pressure  is  about  90  Ibs.  per  sq.  in.,  and  the  heat  of  vaporization, 
in  the  metric  system,  is  about  495  cal.  per  gm. 

9.  How  many  pounds  of  soft  coal  would  be  required  to  change  100  Ibs.  of 
water  at  70°  F.  to  steam  at  212°  F.? 

Assume  that  10  per  cent,  of  the  energy  is  lost  through  incomplete  combus- 
tion, and  that  30  per  cent,  of  the  remaining  heat  escapes  through  the  smoke- 
stack, or  is  lost  by  radiation,  etc.  See  table,  Sec.  181. 

10.  How  high  would  the  energy  obtainable  from  burning  a  ton  of  coal 
raise  a  ton  of  material,  (a)  assuming  12.5  per  cent,  efficiency  for  the  steam 
engine?     (b)  assuming  100  per  cent,  efficiency? 


CHAPTER  XV 
^^i,..-,-  -  VAPORIZATION 

191.  Vaporization  Denned. — Vaporization  is  the  general  term 
applied  to  the  process  of  changing  from  a  liquid  or  solid  to  the 
vapor  state.     Vaporization  takes  place  in  three  different  ways, 
evaporation,  ebullition  (Sec.  192),  and  sublimation.     The  first  two 
refer  to  the  change  from  liquid  to  vapor,  the^last,  from  solid  to 
vapor.     If  aisolid  passes  directly 'iSo  the  vapor  state  without 
first  becoming  aflTJtfid,  it  is  said  to  sublime,  and  the  process  is 
sublimation.     Snow    sublimofr— slowlyV  disappearing  when  per- 
fectly dry  and  far  below  zero.     Other  substances  besides  snow 
sublime;  notably  camphor,  iodine,  and  arsenic. 

In  whatever  manner  the  vaporization  occurs,  it  requires  heat 
energy  to  bring  it  about,  and  when  the  vapor  condenses  an  equal 
amount  of  heat  (the  heat  of  vaporization,  Sec.  182)  is  evolved. 
Hence  a  molecule  must  contain  more  energy  when  in  the  vapor 
state  than  when  in  the  liquid  state,  due,  according  to  the  kinetic 
theory  (Sec.  171),  to  the  greater  rapidity  of  its  to-and-fro  motion. 
The  above  absorption  and  evolution  of  heat  which  accompany 
vaporization  and  condensation,  respectively,  are  of  the  utmost 
importance  in  nature  (Sec.  185)  and  also  commercially.  In 
steam  heating,  the  heat  is  evolved — about  540  calories  for  each 
gram  of  steam  condensed — at  the  place  where  the  condensation 
occurs,  namely,  in  the  radiator.  Note  the  similar  absorption  of 
heat  in  the  melting  of  ice  (utilized  in  the  ice-cream  freezer,  Sec. 
190)  and  the  evolution  of  heat  in  the  freezing  of  water.  Thus, 
vaporization  and  melting  are  heat-absorbing  processes;  while 
the  reverse  changes  of  state,  condensation  and  freezing,  are 
heat-liberating  processes. 

192.  Evaporation  and  Ebullition. — The  heat  energy  of  a  body 
is  supposed  to  be  due  to  its  molecular  motion  (Sec.  160),  which, 
as  the  body  is  heated,  becomes  more  violent.     The  evaporation 
of  a  liquid  may  be  readily  explained  in  accordance  with  this 
theory.     Let  A   (Fig.  128)  be  an  air-tight  cylinder  containing 
some  water  B,  and  provided  with  an  air-tight  piston  P.     Suppose 

260 


VAPORIZATION 


261 


this  piston,  originally  in  contact  with  the  water,  to  be  suddenly 
raised,  thereby  producing  above  the  water  a  vacuum.  As  the 
water  molecules  near  the  surface  of  the  water  move  rapidly  to 
and  fro  some  of  them  escape  into  the  vacuous  space  above,  where 
they  travel  to  and  fro  just  as  do  the  molecules  of  a  gas.  After  a 
considerable  number  of  these  molecules  have  escaped  from  the 
water,  many  of  them  in  their  to-and-fro  motion  will  again  strike 
the  water  and  be  retained.  Thus  we  see  that  there  is  a  continual 
passage  of  these  molecules  from  the  water  to  the  vapor  above, 
and  vice  versa.  The  vapor  above  is  said  to  be  saturated  when,  in 
this  interchange,  equilibrium  has  been  reached;  i.e.,  when  the 
rate  at  which  the  molecules  are  returning  to  the  water  is  equal  to 
the  rate  at  which  they  are  escaping  from  it. 

The  saturated  water  vapor  above  the  water  in  A  exerts  a  pres- 
sure due  to  the  impact  of  its  molecules  against  the  walls,  just  as 


-A 


FIG.  128. 

any  gas  exerts  pressure.  This  vapor  pressure  is  about  1/40 
atmosphere  when  the  water  is  at  room  temperature  and  becomes 
1  atmosphere  when  the  water  and  the  cylinder  are  heated  to  the 
boiling  point. 

Ebullition. — When  water  is  placed  in  an  open  vessel  (C,  Fig. 
128)  evaporation  into  the  air  takes  place  from  the  surface,  as 
already  described  for  vessel  A.  When  heated  to  the  boiling  point 
(D,  Fig.  128),  bubbles  of  vapor  form  at  the  point  of  application 
of  heat  and  rise  to  the  surface,  where  the  vapor  escapes  to  the  air. 
When  vaporization  takes  place  in  this  manner,  i.e.,  by  the  forma- 
tion of  bubbles  within  the  liquid,  it  is  called  Ebullition,  or  boiling; 
while  when  it  takes  place  simply  from  the  surface  of  the  liquid, 
it  is  called  Evaporation. 

As  has  already  been  stated,  the  pressure  of  saturated  water 


262 


MECHANICS  AND  HEAT 


vapor  at  100°  C.  is  one  atmosphere.  This  will  be  evident  from  the 
following  considerations.  In  the  formation  of  the  steam  bubble 
E  below  the  surface  of  the  water  in  the  open  dish  D,  it  is  clear 
that  the  pressure  of  the  vapor  in  the  bubble  must  be  equal  to  the 
atmospheric  pressure  or  it  would  collapse.  Indeed  it  must  be  a 
trifle  greater  than  atmospheric  pressure,  because  the  pressure 
upon  it  is  one  atmosphere  plus  the  slight  pressure  (hdg)  due  to 
the  water  above  it.  We  are  now  prepared  to  accept  the  general 
statement  that  any  liquid  will  boil  in  a  shallow  open  dish  when  it 
reaches  that  temperature  for  which  the  pressure  of  its  saturated 
vapor  is  one  atmosphere.  This  temperature,  known  as  the  boiling 
point  at  atmospheric  pressure  or  simply  the  boiling  point,  differs 
widely  for  the  various  substances. 

193.  Boiling  Point.— Unless  otherwise  stated,  the  Boiling  Point 
is  understood  to  be  that  temperature  at  which  boiling  occurs  at 
Standard  Atmospheric  Pressure  (760  mm.  of  mercury).  For  pure 
liqufds,  this  is  a  perfectly  definite,  sharply  defined  temperature, 
so  definite,  indeed,  that  it  may  be  used  in  identifying  the  sub- 
stance. Thus  if  a  liquid  boils  at  34. °9  we  may  be  fairly  sure  that 
it  is  ether;  at  61°,  chloroform;  at  290°,  glycerine.  The  boiling 
points  for  a  few  substances  are  given  in  the  following  table. 

BOILING  POINTS  AT  ATMOSPHERIC  PRESSURE 


Substance  Temperature 


Substance  Temperature 


Helium  

-267°  C. 

Alcohol  (wood)  .  . 

66°  C. 

Hydrogen  

-253 

Alcohol  (ethyl)... 

78.4 

Nitrogen 

—  194 

Water 

100 

Oxygen 

—  184 

Glycerine 

290 

Carbon  dioxide1.  .  . 
Ammonia  

-  80 
-  38.5 

Mercury  
Sulphur  

357 

448 

Ether  

34.9 

Zinc  

about     930 

Chloroform  

61 

Lead....  

about  1500 

Solutions  of  solids  in  liquids  have  a  higher  boiling  point,  as  well 
as  a  lower  freezing  point  (Sec.  186)  than  the  pure  solvent.  Thus 
a  24  per  cent,  brine,  which  we  have  seen  freezes  at  —22°  C.,  boils 
at  about  107°  C.  The  elevation  of  the  boiling  point  is  approxi- 
mately proportional  to  the  concentration  for  weak  solutions. 
A  24  per  cent,  sugar  solution  boils  at  about  100. °5  C. 

194.  Effect  of  Pressure  on  the  Boiling  Point. — When  a  change 
of  state  is  accompanied  by  an  increase  in  volume,  we  readily  see 

1  Carbon  dioxide  (COz)  sublimes  at  —80°  C.  and  atmospheric  pressure. 
Under  a  pressure  of  5.1  atmospheres  it  melts  and  also  boils  at  —57°  C. 


VAPORIZATION 


263 


that  subjecting  the  substance  to  a  high  pressure  will  oppose  the 
change;  while  if  the  change  of  state  is  accompanied  by  a  decrease 
in  volume,  the  reverse  is  true,  i.e.,  pressure  will  then  aid  the  proc- 
ess. Consequently,  since  water  expands  in  changing  to  either 
ice  or  steam,  subjecting  it  to  high  pressure  makes  it  "harder" 
either  to  freeze  or  boil  it;  i.e.,  pressure  lowers  the  freezing  point, 
(Sec.  186)  and  raises  the  boiling  point.  The  latter  volume  change 
is  vastly  greater  than  the  former;  accordingly  the  corresponding 
temperature  change  is  greater.  Thus,  when  the  pressure  changes 
from  one  atmosphere  to  two,  the  change  of  boiling  point  (21°)  is 
much  greater  than  the  change  of  freezing  point  (O.°0075).  When 
the  steam  gauge  reads  45  Ibs.  per  sq.  in.  or  3  atmospheres,  the 
absolute  steam  pressure  on  the  water  in  the  boiler  is  4  atmospheres 
and  the  temperature  of  the  water  is  144°  C.  When  the  steam 
gauge  reads  200  Ibs.,  a  pressure  sometimes  used,  the  temperature 
of  the  boiler  water  is  194°  C.  On  the  other  hand,  to  make  water 
boil  in  the  receiver  of  an  air  pump  at  room  temperature  (20°),  the 
pressure  must  be  reduced  to  about  1/40  atmosphere.  (See 
table  below.) 

BOILING  POINT  OF  WATER  AT  VARIOUS  PRESSURES  » 


Tempera- 
ture 

Pressure  in 
cm.  of  mer- 
cury 

Tempera- 
ture 

Pressure  in 
atmos- 
pheres 

Tempera- 
ture 

Pressure  in 
atmos- 
pheres 

0°C. 

0.46 

70°  C. 

0.3 

140°  C. 

3.5 

10 

0.92 

80 

0.46 

150 

4.7 

20 

1.74 

90 

0.70 

160 

6.1 

30 

3.15 

100 

1.00 

170 

7.8 

40 

5.49 

110 

1.40 

180 

10.0 

50 

9.20 

121° 

2.00 

190 

12.4 

60 

14.88 

130 

2.67 

200 

15.5 

Franklin's  Experiment  on  Boiling  Point. — Benjamin  Franklin 
discovered  that  if  a  flask  partly  filled  with  water  is  boiled  until 
the  air  is  all  expelled  (Fig.  129,  left  sketch) ,  and  is  then  tightly 
stoppered  and  removed  from  the  flame  (right  sketch),  then  pour- 
ing cold  water  (the  colder  the  better)  upon  the  flask  causes  the 
water  to  boil,  even  after  it  has  cooled  to  about  room  temperature. 
The  explanation  is  simple.  When  the  temperature  of  the  water 
is  50°  C.  the  vapor  pressure  in  the  flask  is  9.2  cm.  of  mercury. 

1  This  is  also  a  table  of  the  saturated  vapor  pressure  of  water  at  various 
temperatures.  (See  close  of  Sec.  192,  also  Sec.  196.) 


264 


MECHANICS  AND  HEAT 


(See  table  above.)  Suppose  that  under  these  conditions  cold 
water  is  poured  upon  the  flask.  This  chilling  of  the  flask  con- 
denses some  of  the  contained  vapor,  thereby  causing  a  slight 
drop  in  pressure,  whereupon  more  water  bursts  into  steam. 
Indeed,  so  long  as  the  temperature  is  50°,  the  vapor  pressure  will 
be  maintained  at  9.2  cm.;  hence  the  colder  the  water  which  is 
poured  on,  the  more  rapid  the  condensation,  and  consequently 
the  more  violent  the  boiling.  The  flask  should  have  a  round 
bottom  or  the  atmospheric  pressure  will  crush  it  when  the  pressure 
within  becomes  low.  Inverting  the  flask  and  placing  the  stopper 
under  water,  as  shown,  precludes  the  possibility  of  air  entering 
the  flask  and  destroying  the  vacuum. 

This  lowering  of  the  boiling  point  as  the  air  pressure  decreases 
is  a  serious  drawback  in  cooking  at  high  altitudes.     At  an  altitude 


FIG.  129. 

of  10,000  ft.  (e.g.,  at  Leadville,  Colorado),  water  boils  at  about 
90°  C.,  and  at  the  summit  of  Pike's  Peak  (alt.  14,000  ft.),  at 
about  85°  C.  At  such  altitudes  it  is  very  difficult  to  cook  (by 
boiling)  certain  articles  of  food,  (e.g.,  beans),  requiring  in  some 
cases  more  than  a  day.  It  will  be  understood  that  when  water 
has  reached  the  boiling  point,  further  application  of  heat  does 
not  cause  any  further  temperature  rise,  but  is  used  in  changing 
the  boiling  water  to  steam.  In  sugar  manufacture,  the  "boiling 
down"  is  done  in  "vacuum  pans"  at  reduced  pressure  and  re- 


VAPORIZATION  265 

duced  temperature  to  avoid  charring  the  sugar.  By  boiling 
substances  in  a  closed  vessel  or  boiler  so  that  the  steam  is  con- 
fined, thereby  raising  the  pressure,  and  consequently  raising  the 
boiling  point,  the  cooking  is  more  quickly  and  more  thoroughly 
done.  This  method  is  used  in  canning  factories. 

Superheating,  Bumping. — After  pure  water  has  boiled  for  some 
time  and  the  air  which  it  contains  has  been  expelled,  it  sometimes 
boils  intermittently  with  almost  explosive  violence  known  as 
"bumping."  A  thermometer  inserted  in  the  water  will  show 
that  the  temperature  just  previous  to  the  "bumping"  is  slightly 
above  normal  boiling  point;  in  other  words  the  water  is  Super- 
heated.  A  few  pieces  of  porous  material  or  a  little  unboiled  water 
added  will  stop  the  bumping.  We  have  seen  (Sec.  186)  that  water 
may  also  be  supercooled  without  freezing.  Dufour  has  shown 
that  water  in  fine  globules  immersed  in  oil  may  remain  liquid 
from  -20°  to  178°  C. 

195.  Geysers. — The  geyser  may  be  described  as  a  great  hot 
spring  which,  at  more  or  less  regular  intervals,  spouts  forth  a 
column  or  jet  of  hot  water.  Geysers  are  found  in  Iceland,  New 
Zealand,  and  Yellowstone  National  Park.  One  of  the  Iceland 
geysers  throws  a  column  of  water  10  ft.  in  diameter  to  a  height 
of  200  ft.  at  intervals  of  about  6  hours.  Grand  Geyser,  of  the 
National  Park,  spouts  to  a  height  of  250  ft.  Old  Faithful,  in 
the  National  Park,  is  noted  for  its  regularity. 

Geysers  owe  their  action  to  the  fact  that  water  under  great 
pressure  must  be  heated  considerably  above  100°  before  it  boils, 
and  perhaps  in  some  cases  also  to  superheating  of  the  lower  parts 
of  the  water  column  just  before  the  eruption  takes  place.  A  deep, 
irregular  passage,  or  "well,"  filled  with  water,  is  heated  at  the 
bottom  by  the  internal  heat  of  the  earth  to  a  temperature  far 
above  the  ordinary  boiling  point  before  the  vapor  pressure  is 
sufficient  to  form  a  bubble.  When  this  temperature  is  reached 
(unless  superheating  occurs)  a  vapor  bubble  forms  and  forces 
the  column  of  water  upward.  At  first  the  water  simply  flows 
away  at  the  top.  This,  however,  reduces  the  pressure  on  the 
vapor  below,  whereupon  it  rapidly  expands,  and  the  highly 
heated  water  below,  now  having  less  pressure  upon  it,  bursts 
into  steam  with  explosive  violence  and  throws  upward  a  column 
of  boiling  water.  This  water,  now  considerably  cooled,  flows 
back  into  the  "  well."  After  a  few  hours  the  water  at  the  bottom 
of  the  well  again  becomes  heated  sufficiently  above  100°  to  form 


266  MECHANICS  AND  HEAT 

steam  bubbles  under  the  high  pressure  to  which  it  is  subjected, 
and  the  geyser  again  "spouts." 

Bunsen,  who  first  explained  the  action  of  the  natural  geyser, 
devised  an  artificial  geyser.  It  consisted  of  a  tin  tube,  say  4  ft. 
in  length  and  4  in.  in  diameter  at  the  lower  end,  tapering  to 
about  1  in.  in  diameter  at  the  top,  with  a  broad  flaring  portion 
above  to  catch  the  column  when  it  spouts.  If  filled  with  water 
and  then  heated  at  the  bottom,  it  spouts  at  fairly  regular  intervals. 
If  constructed  with  thermometers  passing  through  the  walls  of 
the  tube,  it  will  be  found  that  the  thermometers  just  previous  to 
eruption  read  higher  than  100°,  and  that  the  lowest  one  reads 
highest. 

In  the  case  of  steam  boilers  under  high  pressure,  the  water  may 
be  from  50°  to  80°  hotter  than  the  normal  boiling  point,  and  if 
the  boiler  gives  way,  thereby  reducing  the  pressure,  part  of  this 
water  bursts  into  steam.  This  additional  supply  of  steam  no 
doubt  contributes  greatly  to  the  violence  of  boiler  explosions. 

196.  Properties  of  Saturated  Vapor. — If,  after  the  space  above 
the  water  in  A  (Fig.  128)  has  become  filled  with  saturated  vapor, 
the  piston  Pis  suddenly  forced  down,  there  will  then  be  more  mole- 
cules per  unit  volume  of  the  space  than  there  were  before.  Con- 
sequently, the  rate  at  which  the  molecules  return  to  the  water  will 
be  greater  than  before,  and  therefore  greater  than  the  rate  at 
which  they  are  escaping  from  the  water.  In  other  words,  some  of 
the  vapor  condenses  to  water.  This  condensation  takes  place 
very  quickly  and  continues  until  equilibrium  is  restored  and  the 
vapor  is  still  simply  saturated  vapor. 

If,  on  the  other  hand,  the  piston  P  had  been  suddenly  moved 
upward  instead  of  downward,  the  vapor  molecules  in  the  space 
above  the  water,  having  somewhat  more  room  than  before,  would 
not  be  so  closely  crowded  together  and  hence  would  not  return  to 
the  water  in  such  great  numbers  as  before.  In  other  words,  the 
rate  of  escape  of  molecules  from  the  water  would  be  greater  than 
their  rate  of  return;  consequently  the  number  of  molecules  in 
the  space  above  the  water  would  increase  until  equilibrium  was 
reached,  i.e.,  until  the  space  was  again  filled  with  saturated  vapor. 

In  the  case  of  a  saturated  vapor  above  its  liquid,  we  may  con- 
sider that  there  are  two  opposing  tendencies  always  at  work. 
As  the  temperature  of  the  liquid  rises,  the  tendency  of  the  liquid 
to  change  to  vapor  increases,  i.e.,  more  liquid  vaporizes.  The 
effect  of  increasing  the  external  pressure  applied  to  the  vapor  is, 


VAPORIZATION 


267 


on  the  other  hand,  to  tend  to  condense  it  to  the  liquid  state.  At 
all  times,  and  under  all  circumstances,  the  pressure  applied  to 
the  vapor  is  equal  to  the  pressure  exerted  by  the  vapor.  Referring 
to  Fig.  128,  it  may  readily  be  seen  that  if  the  vapor  pressure  act- 
ing upward  upon  P  is  equal  to,  say  5  Ibs.  per  sq.  in.  at  any  instant, 
that  the  downward  pressure  exerted  by  the  piston  upon  the  vapor 
below  it,  is  likewise  5  Ibs.  per  sq.  in.  Of  course  this  would  be 
equally  true  for  any  other  pressure. 


C       D        E       F 


L 


± 


i: 


FIG.  130. 

These  characteristics  of  a  saturated  vapor  above  its  own  liquid 
are  beautifully  illustrated  in  the  following  experiment.  A  ba- 
rometer tube  T  (Fig.  130,  left  sketch)  is  filled  with  mercury, 
stoppered,  and  carefully  inverted  in  a  mercury  "well"  A  about  80 
cm.  deep.  Upon  removing  the  stopper,  the  mercury  runs  out  of 
the  tube,  leaving  the  mercury  level  about  76  cm.  higher  in  the 
tube  than  in  the  well,  as  explained  in  Sec.  136.  Next,  without 
admitting  any  air,  introduce,  by  means  of  an  ink  filler,  sufficient 
ether  to  make  about  1  cm.  depth  in  the  tube.  This  ether  rises 


268  MECHANICS  AND  HEAT 

and  quickly  evaporates,  until  the  upper  part  of  the  tube  is  filled 
with  its  saturated  vapor,  whose  pressure  at  room  temperature  is 
about  2/3  atmosphere.  Consequently,  the  mercury  drops  until 
it  is  about  25  cm.  (1/3  of  76)  above  that  in  the  well. 

Now,  as  the  tube  is  quickly  moved  upward  more  ether  evapo- 
rates, maintaining  the  pressure  of  the  saturated  vapor  con- 
stantly at  2/3  atmosphere,  as  evidenced  by  the  fact  that  the  level 
of  the  mercury  still  remains  25  cm.  above  that  in  the  well.  If  the 
tube  is  suddenly  forced  downward,  some  ether  vapor  condenses, 
and  the  mercury  still  remains  at  the  same  25-cm.  level.  After  the 
tube  has  been  raised  high  enough  that  all  of  the  ether  is  evapor- 
ated, further  raising  it  causes  the  pressure  of  the  vapor  to  de- 
crease (in  accordance  with  Boyle's  Law),  as  shown  by  the  fact 
that  the  level  of  the  mercury  in  the  tube  then  rises. 

Finally,  if  the  tube  and  contents  are  heated  to  34.9°  C.,  the 
boiling  point  for  ether,  its  saturated  vapor  produces  a  pressure  of 
one  atmosphere,  and  the  mercury  within  and  without  the  tube 
comes  to  the  same  level,  and  remains  at  the  same  level  though  the 
tube  be  again  raised  and  lowered.  If  the  tube  is  severely  chilled, 
the  mercury  rises  considerably  higher  than  25  cm.  This  shows 
that  the  pressure  of  the  saturated  vapor,  or  the  pressure  at  which 
boiling  occurs,  rises  rapidly  with  the  temperature.  (See  table 
for  Water,  Sec.  194.) 

Saturated  Vapor  Pressure  of  Different  Liquids. — The  pressure  of 
the  saturated  vapor  of  liquids  varies  greatly  for  the  different 
liquids,  as  shown  by  the  experiment  illustrated  in  Fig.  130  (right 
sketch).  The  four  tubes  C,  D,  E,  and  F  are  filled  with  mercury 
and  are  then  inverted  in  the  mercury  trough  G.  The  mercury 
then  stands  at  a  height  of  76  cm.  in  each  tube.  If,  now,  a  little 
alcohol  is  introduced  into  D,  some  chloroform  into  E,  and  some 
ether  into  F,  it  will  be  found  that  the  mercury  level  lowers  by  the 
amounts  hi,  hz,  and  h3,  respectively.  The  value  of  hi  is  4.4  cm., 
which  shows  that  at  room  temperature  the  pressure  of  the  sat- 
urated vapor  of  alcohol  is  equal  to  4.4  cm.  of  mercury.  Since 
h2/hi  =  4:  (approx.),  we  see  that  at  room  temperature  the  pressure 
of  the  saturated  vapor  is  about  4  times  as  great  for  chloroform 
as  for  alcohol. 

197.  Cooling  Effect  of  Evaporation. — If  the  hand  is  mois- 
tened with  ether,  alcohol,  gasoline,  or  any  other  liquid  that  evapo- 
rates quickly,  a  decided  cooling  effect  is  produced.  Water  pro- 
duces a  similar  but  less  marked  effect.  We  have  seen  that  it 


VAPORIZATION  269 

requires  536.5  calories  to  change  a  gram  of  boiling  water  to  steam. 
When  water  is  evaporated  at  ordinary  temperatures  it  requires 
somewhat  more  than  this,  about  600  calories.  If  this  heat  is  not 
supplied  by  a  burner  or  some  other  external  source,  it  must  come 
from  the  remaining  water  and  the  containing  vessel,  thereby 
cooling  them  below  room  temperature. 

There  are  two  factors  which  determine  the  magnitude  of  the 
cooling  effect  produced  by  the  evaporation  of  a  liquid.  One  of 
these  is  the  volatility  of  the  liquid;  the  other,  the  value  of  its 
heat  of  vaporization.  From  the  table  (Sec.  182)  we  see  that  the 
heat  of  vaporization  is  about  3  times  as  great  for  water  as  for 
alcohol.  Consequently,  if  alcohol  evaporated  3  times  as  fast  as 
water  under  like  conditions,  then  alcohol  and  water  would  pro- 
duce about  equally  pronounced  cooling  effects.  Alcohol,  how- 
ever, evaporates  much  more  than  3  times  as  fast  as  water,  and 
therefore  gives  greater  cooling  effect,  as  observed. 

If  three  open  vessels  contain  alcohol,  chloroform,  and  ether, 
respectively,  it  will  be  found  that  a  thermometer  placed  in  the 
one  containing  alcohol  shows  a  temperature  slightly  lower  than 
room  temperature;  while  the  one  in  chloroform  reads  still  lower, 
and  the  one  in  ether  the  lowest  of  all.  A  thermometer  placed  in 
water  would  read  almost  exactly  room  temperature.  The  main 
reason  for  this  difference  is  the  different  rates  at  which  these 
liquids  evaporate,  although,  as  just  stated,  the  value  of  the  heat 
of  vaporization  is  also  a  determining  factor.  Ether,  being  by  far 
the  most  volatile  of  the  three,  gives  the  greatest  cooling  effect. 
Observe  that  the  more  volatile  liquids  are  those  having  a  low 
boiling  point,  and  consequently  a  high  vapor  pressure  at  room 
temperature.  In  some  minor  surgical  operations  the  requisite 
numbness  is  produced  by  the  chilling  effect  of  a  spray  of  very 
volatile  liquid.  Other  practical  uses  of  the  cooling  effect  of 
evaporation  are  discussed  in  Sees.  198,  199,  and  200.  The 
converse,  or  the  heating  effect  due  to  condensation,  is  utilized  in 
all  heating  by  steam  (Sec.  191),  and  it  also  plays  an  important  role 
in  influencing  weather  conditions. 

198.  The  Wet-and -dry-bulb  Hygrometer. — The  'cooling 
effect  of  evaporation  is  employed  in  the  wet-and-dry-bulb 
hygrometer,  used  in  determining  the  amount  of  moisture  in  the 
atmosphere.  It  consists  of  two  ordinary  thermometers  which 
are  just  alike  except  that  a  piece  of  muslin  is  tied  about  the  bulb 
of  one.  The  muslin  is  in  contact  with  a  wick,  the  lower  end  of 


270  MECHANICS  AND  HEAT 

which  is  in  a  vessel  of  water.  By  virtue  of  the  capillary  action  of 
the  wick  and  muslin,  the  bulb  is  kept  moist.  This  moisture 
evaporating  from  the  bulb  cools  it,  causing  this  thermometer  to 
read  several  degrees  lower  than  the  other  one. 

If  the  air  is  very  dry,  this  evaporation  will  be  rapid  and  the 
difference  between  the  readings  of  the  two  thermometers  will  be 
large;  whereas  if  the  air  is  almost  saturated  with  moisture,  the 
evaporation  will  be  slow  and  the  two  thermometer  readings  will 
differ  but  slightly.  Consequently,  if  the  two  readings  differ  but 
little,  rain  or  other  precipitation  may  be  expected.  The  method 
of  finding  the  amount  of  water  vapor  in  the  air  by  means  of  these 
thermometer  readings,  is  discussed  in  a  subsequent  chapter. 

As  a  mass  of  air  m  comes  into  contact  with  the  wet  (colder)  bulb  it 
gives  heat  to  the  bulb,  and  as  it  absorbs  moisture  from  the  bulb  it  also 
takes  heat  from  it.  A  few  moments  after  the  apparatus  is  set  up,  equi- 
librium is  reached,  as  shown  by  the  fact  that  the  temperature  of  the  wet 
bulb  is  constant.  It  is  then  known  that  the  amounts  of  heat  "given" 
and  "taken"  by  the  bulb  are  equal.  This  fact  is  utilized  in  the  deriva- 
tion of  certain  theoretical  formulas  for  computing  the  amount  of 
moisture  in  the  air  directly  from  the  two  thermometer  readings.  The 
practical  method,  however,  is  to  use  tables  (Sec.  222)  compiled  from 
experiments. 

199.  Cooling  Effect  due  to  Evaporation  of  Liquid  Carbon 
Dioxide. — Carbon  dioxide  (COa)  is  a  gas  at  ordinary  tempera- 
tures and  pressures,  but  if  cooled  to  a  low  temperature  and  then 
subjected  to  high  pressure  it  changes  to  the  liquid  state.  If 
the  pressure  is  reduced  it  quickly  changes  back  to  the  vapor  state. 
We  have  seen  that  the  pressure  of  water  vapor  is  about  1/40 
atmosphere  at  room  temperature.  Liquid  carbon  dioxide  is  so 
extremely  volatile,  that  is,  it  has  so  great  a  tendency  to  change 
to  the  vapor  state,  that  its  vapor  pressure  at  room  temperature 
has  the  enormous  value  of  60  atmospheres.  It  follows  then,  that 
when  an  air-tight  vessel  is  partly  filled  with  liquid  carbon  dioxide 
at  room  temperature,  a  portion  of  it  quickly  changes  to  vapor 
until  the  pressure  in  the  space  above  the  liquid  becomes  60  atmos- 
pheres. Carbon  dioxide  is  shipped  and  kept  in  strong  sealed 
iron  tanks  to  be  used  for  charging  soda  fountains,  etc. 

If  such  a  tank  is  inverted  (Fig.  131)  and  the  valve  is  opened, 
a  stream  of  liquid  carbon  dioxide  is  forced  out  by  the  60-atmos- 
phere  pressure  of  the  vapor  within.  As  soon  as  this  liquid  carbon 
dioxide  escapes  to  the  air,  where  the  pressure  is  only  one  atmos- 
phere, it  changes  almost  instantly  to  vapor,  and  takes  from  the 


VAPORIZATION 


271 


air,  from  the  nozzle,  and  from  the  remaining  liquid,  its  heat  of 
vaporization,  about  40  calories  per  gram  at  room  temperature. 
This  abstraction  of  heat  chills  the  nozzle  to  such  an  extent  that 
the  moisture  of  the  air  rapidly  condenses  upon  it  as  a  frosty 
coating.  It  also  chills,  in  fact  freezes,  part  of  the  liquid  jet  of 
carbon  dioxide,  forming  carbon  dioxide  "snow."  This  snow  is 
so  cold  ( —80°  C.)  that  mercury  surrounded  by  it  quickly  freezes. 

200.  Refrigeration  and  Ice 
Manufacture  by  the  Ammonia 
Process. — There  are  several  sys- 
tems or  methods  of  ice  manufac- 
ture, in  all  of  which,  however,  the 
chilling  effect  is  produced  by  the 
heat  absorption  (due  to  heat  of 
vaporization)  that  accompanies 
the  vaporization  of  a  volatile 
liquid.  The  most  important  of 
these  liquids  are  ammonia  (NH3) 
and  carbon  dioxide  (CO2).  Econ- 
omy demands  that  the  vapor  be 
condensed  again  to  a  liquid,  in 
order  to  use  the  same  liquid  re- 
peatedly. 

In  the  Compression  System,  the 
vapor  is  compressed  by  means  of 
an  air  pump  until  it  becomes  a 

liquid.  The  heat  evolved  in  this  process  (heat  of  vaporization) 
is  disposed  of  usually  by  flowing  water,  and  the  cooled  liquid 
(e.g.,  ammonia)  is  again  allowed  to  evaporate.  Thus  the  cycle, 
consisting  of  evaporation  accompanied  by  heat  absorption,  and 
condensation  to  liquid  accompanied  by  heat  evolution,  is  repeated 
indefinitely.  Since  the  former  occurs  in  pipes  in  the  ice  tank 
(freezing  tank),  we  see  that  the  heat  is  literally  pumped  from  the 
freezing  tank  to  the  flowing  water. 

Ammonia  is  a  substance  admirably  adapted  to  use  in  this  way. 
Its  heat  of  vaporization  is  fairly  large  (295  cal.  per  gm.),  and  it  is 
very  volatile— that  is,  it  evaporates  very  quickly,  its  vapor 
pressure  at  room  temperature  being  about  10  atmospheres.  At 
—38°. 5  C.  its  vapor  pressure  is  one  atmosphere;  hence  it  would 
boil  in  an  open  vessel  at  that  low  temperature.  The  liquid  com- 
monly called  ammonia  is  simply  water  containing  ammonia  gas 


FIG.  131. 


272 


MECHANICS  AND  HEAT 


which  it  readily  absorbs.  Ammonia  is  a  gas  at  ordinary  tempera- 
tures, but  when  cooled  and  subjected  to  several  atmospheres' 
pressure  it  changes  to  a  liquid.  If  carbon  dioxide  is  used  instead 
of  ammonia,  the  cost  of  manufacturing  the  ice  is  somewhat 
greater.  The  greater  compactness  of  the  apparatus,  however, 
coupled  with  the  fact  that  in  case  of  accidental  bursting  of  the 
pipes,  carbon  dioxide  is  much  less  dangerous  than  ammonia,  has 
resulted  in  its  adoption  on  ships. 

The  essentials  of  the  Ammonia  Refrigerating  apparatus  are 
shown  diagrammatically  in  Fig.  132.  A  is  the  cooling  tank  which 
receives  a  continual  supply  of  cold  water  through  pipe  c.;B  is  an 
air  pump;  C  is  a  freezing  tank  filled  with  brine;  D  is  a  pipe  filled 
with  liquid  ammonia;  and  E  is  a  pipe  filled  with  ammonia  vapor. 


FIG.  132. 

If  valve  F  were  slightly  opened,  liquid  ammonia  would  enter  E 
and  evaporate  until  the  pressure  in  E  was  equal  to  the  vapor 
pressure  of  ammonia  at  room  temperature  or  about  10  atmos- 
pheres. Whereupon  evaporation,  and  therefore  all  cooling  action , 
would  cease.  If,  however,  the  pump  is  operated,  ammonia  gas 
is  withdrawn  from  E  through  valve  a  and  is  then  forced  into  pipe 
D  through  valve  6  under  sufficient  pressure  to  liquefy  it.  This 
constant  withdrawal  of  ammonia  gas  from  pipe  E  permits  more 
liquid  ammonia  to  enter  through  F  and  evaporate.  The  am- 


VAPORIZATION  273 

monia,  as  it  evaporates  in  E,  withdraws  from  E  and  from  the  sur- 
rounding brine  its  heat  of  vaporization  (about  300  cal.  per  gm.); 
while  each  gram  of  gas  that  is  condensed  to  a  liquid  in  D  imparts 
to  D  and  to  its  surroundings  about  300  calories.  Thus  we  see 
that  heat  is  withdrawn  from  the  very  cold  brine  in  C  and  imparted 
to  the  much  warmer  water  in  A.  This  action  continues  so  long 
as  the  pump  is  operated.  Brine  is  used  in  C  because  it  may  be 
cooled  far  below  zero  without  freezing. 

The  Refrigerator  Room. — The  cold  brine  from  C  may  be  pumped 
through  d  into  the  pipes  in  the  refrigerator  room  and  then  back 
through  pipe  e  to  the  tank.  The  brine  as  it  returns  is  not  so  cold 
as  before,  having  abstracted  some  heat  from  the  refrigerator 
room.  This  heat  it  now  imparts  to  pipe  E.  Thus,  through  the 
circulation  of  the  brine,  heat  is  carried  from  the  cooling  room  to 
the  tank  C,  and  we  have  just  seen  that  due  to  the  circulation  of 
the  ammonia,  heat  is  carried  from  the  brine  tank  C  to  the  water 
tank  A. 

The  pipe  E,  instead  of  passing  into  the  brine,  may  pass  back 
and  forth  in  the  refrigerator  room.  The  stifling  ammonia  vapor, 
which  rapidly  fills  the  room,  in  case  of  the  leaking  or  bursting  of 
an  ammonia  pipe,  makes  this  method  dangerous. 

In  the  Can  System  of  ice  manufacture,  the  cans  of  water  to  be 
frozen  are  placed  in  the  brine  in  C,  and  left  there  40  or  50  hours 
as  required.  In  the  Plate  System,  the  pipe  E  passes  back  and  forth 
on  one  face  of  a  large  metal  plate,  chilling  it  and  forming  a  sheet 
of  ice  of  any  desired  thickness  upon  the  other  face,  which  is  in 
contact  with  water.  For  every  8  or  10  tons  of  ice  manufactured, 
the  engine  that  operates  the  pump  uses  about  one  ton  of  coal. 

Observe  that  in  "pumping"  the  heat,  as  we  may  say,  from  the 
cold  freezing  tank  to  the  much  warmer  flowing  water,  we  are 
causing  the  heat  to  flow  "uphill,"  so  to  speak;  for  heat. of  itself 
always  tends  to  flow  from  hotter  to  colder  bodies,  that  is,  "down- 
hill." Observe  also  that  it  takes  external  applied  energy  of  the 
steam  engine  that  operates  the  pump  to  cause  this  "uphill"  flow 
of  heat. 

201.  Critical  Temperature  and  Critical  Pressure.— In  1869, 
Dr.  Andrews  performed  at  Glasgow  his  classical  experiments  on 
carbon  dioxide.  He  found  that  when  some  of  this  gas,  confined 
in  a  compression  cylinder  at  a  temperature  of  about  32°  or  33°  C., 
had  the  pressure  upon  it  changed  from  say  70  atmospheres  to:  80 
atmospheres,  then  the  volume  decreased,  not  by  1/8  as  required 


274 


MECHANICS  AND  HEAT 


by  Boyle's  Law  (Sec.  139),  but  much  more  than  this.  He  also 
found  that  carbon  dioxide  gas  cannot  be  changed  to  the  liquid 
state  by  pressure,  however  great,  if  its  temperature  is  above  31°  C. 
This  temperature  (31°)  is  called  the  Critical  Temperature  for 
carbon  dioxide. 

If  carbon  dioxide  gas  is  at  its  critical  temperature,  it  requires 
73  atmospheres'  pressure  to  change  it  to  the  liquid  state.  This 
pressure  is  called  the  Critical  Pressure  for  carbon  dioxide.  If 
the  temperature  of  any  gas  is  several  degrees  lower  than  its 
critical  temperature,  then  the  pressure  required  to  change  it  to 
the  liquid  state  is  considerably  less  than  the  critical  pressure. 
Below  is  given  a  table  of  critical  temperatures  and  critical  pres- 
sures for  a  few  gases. 

CRITICAL  TEMPERATURES  AND  CRITICAL  PRESSURES  FOR  A  FEW 
SUBSTANCES 


Substance 

Critical  temperature 

Critical  pressure  in 
atmospheres 

Hydrogen1  (H)  
Nitrogen  (N)  

-241°  C. 
-146 

14 
34 

Air  (O  and  N)  

-140 

39 

Oxygen  (O) 

—  118 

50 

Ethylene  (C2H4)  
Carbon  dioxide  (CCh)  
Ammonia  (NHs)  

10 
30.92 
130 

52 
73 
115 

Water  vapor  (H,O)  

364 

194.6 

202.  Isothermals  for  Carbon  Dioxide. — In  Fig.  134,  the  isothermals 
which  Andrews  determined  for  carbon  dioxide  are  shown.  For  the 
meaning  of  isothermals  and  the  method  of  obtaining  them,  the  student 
is  referred  to  "Isothermals  for  Air"  (Sec.  140). 

The  essential  parts  of  the  apparatus  used  by  Andrews  are  shown  in 
section  in  Fig.  133.  A  glass  tube  A  about  2.5  mm.  in  diameter,  terminat- 
ing in  a  fine  capillary  tube  above,  was  filled  with  carbon  dioxide  gas  and 
plugged  with  a  piston  of  mercury  a.  This  tube  was  next  slipped  into 
the  cap  C  of  the  compression  chamber  D.  A  similar  tube  B,  filled  with 
air,  and  likewise  stoppered  with  mercury,  was  placed  in  the  compression 
chamber  E. 

As  S  was  screwed  into  the  compression  chamber  D,  the  pressure 

1  The  values  —  234.°5  C.  and  20  atmospheres,  sometimes  given  as  the 
critical  temperature  and  critical  pressure,  respectively,  for  hydrogen,  are 
incorrect;  the  first,  because  of  extrapolation  error  in  the  readings  of  the  resist- 
ance thermometer,  the  second,  because  of  manometer  error  in  the  original 
determination. 


VAPORIZATION 


275 


on  the  water  in  the  two  chambers,  and  consequently  the  pressure  on 
the  mercury  and  gas  in  the  two  tubes  A  and  B,  could  be  increased  as 
desired.  Of  course,  as  the  pressure  was  increased,  the  mercury  rose 
higher  and  higher  in  tubes  A  and  B  to,  say,  mi  and  m2.  Knowing  the 
original  volume  of  air  in  B  and  also  the  bore  of  the  capillary  portion  of 
tube  B,  the  pressure  in  the  chamber  could  be  determined.  Thus,  if 
the  volume  of  air  in  tube  B  above  m2  were  1/50  of  the  original  volume, 
then  the  pressure  in  both  chambers  would  be  approximately  50  atmos- 
pheres. At  such  pressures  there  is  a  deviation  from  Boyle's  law,  which 
was  taken  into  account  and  corrected 
for.  Knowing  the  bore  of  A ,  the  volume 
of  carbon  dioxide  above  mi  could  be 
found. 

Plotting  the  values  of  the  volumes  so 
found  as  abscissa,  with  the  correspond- 
ing pressures  as  ordinates,  when  the 
temperature  of  the  apparatus  was  48.°1 
C.,  the  isothermal  marked  48.°1  (Fig. 
134)  was  obtained.  The  form  of  the 
48. °1  isothermal  shows  that  at  this  tem- 
perature the  carbon  dioxide  vapor  fol- 
lowed Boyle's  law,  at  least  roughly. 

When,  however,  the  experiment  was 
repeated  with  the  apparatus  at  the  tem- 
perature 31.°  1  C.,  it  was  found  that  when 
the  pressure  was  somewhat  above  70 
atmospheres  (point  a  on  the  31.°1  isother- 
mal) a  slight  increase  in  pressure  caused  a 
very  great  decrease  in  volume,  as  shown  by 
a  considerable  rise  in  mi.  As  the  pres- 
sure was  increased  slightly  above  75  at- 
mospheres, as  represented  by  point  6  on 

the  curve,  a  further  slight  reduction  of  volume  was  accompanied  by  a 
comparatively  great  increase  in  pressure,  as  shown  by  the  fact  that  the 
portion  be  of  the  isothermal  is  nearly  vertical.  Note  also  that  the  por- 
tion ab  of  the  isothermal  is  nearly  horizontal. 

If  the  experiment  were  again  repeated  at,  say  30°  C.,  then  as  the  pres- 
sure reached  about  70  atmospheres,  liquid  carbon  dioxide  would  collect 
on  mi,  and  this  liquid  would  be  seen  to  have  a  sharply  defined  meniscus 
separating  it  from  the  vapor  above.  At  31.°1  no  such  meniscus  appears. 
The  limiting  temperature  (30.°92  C.)  at  which  the  meniscus  just  fails 
to  appear  under  increasing  pressure,  is  called  the  Critical  Temperature. 

Let  us  now  discuss  the  21.°5  isothermal,  which  isothermal  was  deter- 
mined by  keeping  the  apparatus  at  21.°5  while  increasing  the  pressure. 
As  the  volume  was  decreased  from  that  represented  by  point  A  to  that 


FIG.    133. 


276 


MECHANICS  AND  HEAT 


represented  by  point  B,  the  pressure  increased  from  about  50  atmospheres 
to  60.  Now  as  S  was  screwed  farther  into  the  chamber,  the  volume 
decreased  from  point  B  to  point  C  with  practically  no  increase  in  pressure 
(note  that  BC  is  practically  horizontal).  During  this  change  the  satu- 
rated carbon  dioxide  vapor  was  changing  to  the  liquid  state,  as  shown 
by  the  fact  that  the  liquid  carbon  dioxide  resting  on  mi  could  be  seen  to 
be  increasing.  At  C  the  gas  had  all  been  changed  to  liquid  carbon 
dioxide,  and  since  liquids  are  almost  incompressible,  a  very  slight  com- 
pression, i.e.,  a  very  slight  rising  of  meniscus  mi,  was  accompanied  by  a 
very  great  increase  of  pressure,  as  evidenced  by  the  nearly  vertical  direc- 
tion of  CD. 

It  will  be  observed,  that  while  the  volume  is  reduced  from  that  rep- 
resented by  point  B  to  that  represented  by  point  C,  the  carbon  dioxide 
100 


Volume 

FIG.  134. 

is  changing  to  the  liquid  state,  and  therefore  gaseous  and  liquid  carbon 
dioxide  coexist  in  tube  A.  Likewise  at  13.°1  the  two  states,  or  phases, 
coexist  from  B^  to  Ci,  while  if  the  temperature  were,  say  28°  C.,  the  two 
phases  would  coexist  for  volumes  between  B2  and  C2.  Accordingly,  the 
region  within  the  dotted  curve  through  B,  BI,  Bz,  C,  Ci,  C2,  etc.,  represents 
on  the  diagram  all  possible  corresponding  values  of  pressure,  volume, 
and  temperature  at  which  the  two  phases  may  coexist.  Thus,  if  the 
state  of  the  carbon  dioxide  (temperature,  pressure,  and  volume)  is  rep- 
resented by  a  point  anywhere  to  the  right,  or  to  the  right  and  above 
this  dotted  curve,  only  the  gaseous  phase  exists;  to  the  left,  only  the 
liquid  phase.  We  may  now  define  the  Critical  Temperature  of  any  sub- 
stance as  the  highest  temperature  at  which  the  liquid  and  gaseous  phases 
of  that  substance  can  coexist. 


VAPORIZATION  277 

This  definition  suggests  the  following  simple  method  of  determining 
critical  temperatures.  A  thick-walled  glass  tube  is  partly  (say  1/4) 
filled  with  the  liquid,  e.g.  water,  the  space  above  being  a  vacuum,  or 
rather,  a  space  containing  saturated  water  vapor.  The  tube  is  then 
heated  until  the  meniscus  disappears.  The  temperature  at  which  the 
meniscus  disappears  is  the  critical  temperature  (364°  C.  for  water),  and 
the  pressure  then  tending  to  burst  the  tube,  is  termed  the  critical  pressure. 
It  will  be  noted  that  as  the  water  is  heated,  its  vapor  pressure  becomes 
greater,  finally  producing  the  critical  pressure  (194.6  atmospheres) 
when  heated  to  the  critical  temperature. 

The  Distinction  between  a  Vapor  and  a  Gas. — When  a  gas  is  cooled 
below  its  critical  temperature  it  becomes  a  vapor.  Conversely,  when  a 
vapor  is  heated  above  its  critical  temperature  it  becomes  a  gas.  A 
vapor  and  its  liquid  often  coexist;  a  gas  and  its  liquid,  never. 

203.  The  Joule-Thomson  Experiment. — In  1852,  Joule  and  Thomson 
(Lord  Kelvin)  performed  their  celebrated  "Porous  Plug"  experiment. 
They  forced  various  gases  under  high  pressure  through  a  plug  of  cotton 
or  silk  into  a  space  at  atmospheric  pressure.  In  every  case,  except  when 


FIG.  135. 

hydrogen  was  used,  the  gas  was  cooler  after  passing  through  the  plug 
than  it  was  before.  Hydrogen,  on  the  contrary,  showed  a  slight  rise 
in  temperature.  We  may  note,  however,  that  at  very  low  temperatures 
(below— 80°  C.)  hydrogen  also  experiences  a  cooling  effect. 

The  principle  involved  in  this  experiment  will  be  explained  in  connec- 
tion with  Fig.  135.  Let  P  be  a  stationary  porous  plug  in  a  cylinder  con- 
taining two  pistons  C  and  D.  Let  piston  C,  as  it  moves  (slowly)  from 
Ai  to  BI  against  a  high  pressure  plt  force  the  gas  of  volume  Vi  through 
the  plug,  and  let  this  gas  push  the  piston  D  from  A2  to  52,  and  let  it 
have  the  new  volume  Vz  and  the  new  pressure  pz  (1  atmosphere). 
Now,  from  the  proof  given  in  Sec.  156,  we  see  that  piston  C  does  the  work 
p\Vi  upon  the  gas  in  forcing  it  through  the  plug;  while  the  work  done 
by  the  gas  in  forcing  D  from  A*  to  B2  is  p2V2.  Accordingly,  if  p2Vz  = 
piVi,  i.e.,  if  the  work  done  by  the  gas  is  equal  to  the  work  done  upon  it, 
then  the  gas  should  (on  this  score  at  least)  be  neither  heated  nor  cooled 
by  its  passage  through  the  plug.  All  gases,  however,  deviate  from 
Boyle's  law,  and  for  all  but  hydrogen  the  product  pV  at  ordinary  tem- 
peratures increases  as  p  decreases.  Hence  here  p2V2>piFi  (>  =  is 
greater  than),  which  means  that  the  work  done  by  the  gas  (which  tends 
to  cool  it)  exceeds  the  work  done  upon  the  gas  (which  tends  to  heat  it). 


278  MECHANICS  AND  HEAT 

As  a  result,  then,  the  gas  is  either  cooled  or  else  it  abstracts  heat  from  the 
piston,  or  both. 

Cooling  Effect  of  Internal  Work. — From  the  known  deviation  from 
Boyle's  law  exhibited  by  air,  it  can  be  shown  that  the  temperature  of 
the  air  in  passing  through  the  plug  should  drop  about  O.°l  C.  for  each 
atmosphere  difference  in  pressure  between  pi  and  p2.  Thomson  and 
Joule  found  a  difference  of  nearly  1°  C.  per  atmosphere.  This  addi- 
tional cooling  effect  is  attributed  to  the  work  done  against  intermolec- 
ular  attraction  (internal  work  done)  when  a  gas  expands.  The  work 
done  by  the  gas  in  expanding  is  due,  then,  in  part  to  the  resulting  in- 
crease in  pV  (deviating  from  Boyle's  law),  and  in  part  to  the  work  done 
against  intermodular  attraction  in  increasing  the  average  distance  be- 
tween its  molecules.  Both  of  these  effects,  though  small,  are  more 
marked  at  low  temperatures,  and  by  an  ingenious  but  simple  arrange- 
ment for  securing  a  cumulative  effect,  Linde  has  employed  this  prin- 
ciple in  liquefying  air  and  other  gases  (Sec.  206).  In  Linde's  apparatus, 
the  gas  passes  through  a  small  opening  in  a  valve  instead  of  through  a 
porous  plug. 

204.  Liquefaction  of  Gases. — -About  the  beginning  of  the 
present  century,  one  after  another  of  the  so-called  permanent 
gases  were  liquefied,  until  now  there  is  no  gas  known  that  has  not 
been  liquefied.  Indeed  most  of  them  have  not  only  been  lique- 
fied, but  also  frozen. 

In  1823,  the  great  experimenter  Faraday  liquefied  chlorine  and 
several  other  gases  with  a  very  simple  piece  of  apparatus.  The 
chemical  containing  the  gas  to  be  liquefied  was  placed  in  one  end 
of  a  bent  tube,  the  other  end  of  which  was  placed  in  a  freezing 
mixture  producing  a  temperature  lower  than  the  critical  tempera- 
ture of  the  gas.  The  end  of  the  tube  containing  the  chemical 
was  next  heated  until  the  gas  was  given  off  in  sufficient  quantity 
to  produce  the  requisite  pressure  to  liquefy  it  in  the  cold  end  of 
the  tube. 

In  1877,  Pictet  and  Cailletet  independently  succeeded  in 
liquefying  oxygen.  Later  Professor  Dewar  and  others  liquefied 
air,  and  in  1893  Dewar  froze  some  air.  A  few  years  later  (1897) 
he  liquefied  and  also  (1899)  froze  some  hydrogen.  Subsequently 
(1903)  he  produced  liquid  helium,  a  substance  that  boils  at  6° 
on  the  absolute  scale  or  at  —  267°  C.  He  also  invented  the  Dewar 
flask  (Sec.  206),  in  which  to  keep  these  liquids. 

In  liquefying  air  and  other  gases  having  low  critical  tempera- 
tures, the  great  difficulty  encountered  is  in  the  production  and 
maintenance  of  such  low  temperatures.  To  accomplish  this,  the 


VAPORIZATION 


279 


cooling  effect  of  the  evaporation  of  a  liquid  and  the  cooling  effect 
produced  when  a  gas  expands  (Sec.  178)  have  both  been  utilized. 

There  are  two  distinctly  different  methods  of  liquefying  air, 
known  as  the  "Cascade"  or  Series  Method,  due  to  Raoult  Pictet 
(Sec.  205),  and  the  "Regenerative  Method,"  due  to  Linde  and 
others  (Sec.  206). 

205.  The  Cascade  Method  of  Liquefying  Gases.— In  Fig.  136  is 
shown  a  diagrammatic  sketch  of  the  apparatus  of  Pictet,  as  modified 
and  used  with  great  success  in  the  latter  part  of  the  19th  Century  by 
Dewar,  Olszewski,  and  others.  It  consists  of  three  vessels  A,  B,  and  C, 
the  two  air  pumps  D  and  E,  and  the  carbon  dioxide  tube  F,  together 
with  the  connecting  pipes  as  shown. 

The  pump  D  forces  ethylene  through  pipe  K,  valve  G,  and  pipe  M 
into  the  vessel  B  from  which  vessel  the  ethylene  (now  in  the  vapor  state) 
returns  to  the  pump  through  pipe  N.  Pump  E  maintains  a  similar 


FIG    136 

counterclockwise  circulation  of  air  through  L,  H,  0,  C,  and  P,  as  is 
indicated  by  the  arrows. 

The  vaporization  of  the  carbon  dioxide  in  A  produces  a  temperature 
of  —80°  C.  (Sec.  199).  This  cold  gas,  coming  in  contact  with  the  spiral 
pipe  K  (shown  straight  to  avoid  confusion),  cools  it  enough  that  the 
ethylene  within  it  liquefies  under  the  high  pressure  to  which  it  is  sub- 
jected. As  this  cold  liquid  ethylene  vaporizes  at  M,  it  cools  the  air  in 
L  to  such  an  extent  that  it  in  turn  liquefies  under  the  high  pressure  pro- 
duced by  pump  E.  As  this  liquid  air  passes  through  valve  H  and 
vaporizes  in  C,  it  produces  an  extremely  low  temperature.  As  pointed 
out  in  the  discussion  of  the  ammonia  refrigerating  apparatus,  the  main- 
tenance of  a  partial  vacuum  into  which  the  liquid  may  vaporize,  as  in 
B  and  C,  causes  more  rapid  vaporization,  and  therefore  enhances  the 
chilling  effect.  The  liquid  air  may  be  withdrawn  at  I,  and  fresh  air  may 
be  admitted  at  J  to  replenish  the  supply. 


280 


MECHANICS  AND  HEAT 


In  liquefying  air  by  this  method,  it  is  necessary  to  use  ethylene,  or 
some  other  intermediate  liquid  which  produces  a  very  low  temperature 
when  vaporized.  For  if  L  simply  passed  through  vessel  A,  no  pressure, 
however  great,  would  liquefy  the  air  within  it,  since  —80°  C.  is  above  the 
critical  temperature  for  air.  Gases  have,  however,  been  liquefied  when 
at  temperatures  considerably  above  the  critical  temperatures,  by  sub- 
jecting them  to  enormous  pressures  and  then  suddenly  relieving  the 
pressure. 

206.  The  Regenerative  Method  of  Liquefying  Gases. — The 
regenerative  method  of  liquefying  gases  employs  the  principle 
(established  by  Thomson  and  Joule,  Sec.  203)  that  a  gas  is  chilled 

200  Atmospheres  Cold 


as  it  escapes  through  an  orifice  from  a  region  of  high  pressure  to 
a  region  of  low  pressure.  This  method  has  made  possible  the 
liquefaction  of  every  known  gas,  and  also  the  production  of  liquid 
air  in  large  quantities  and  at  a  greatly  reduced  cost.  From  about 
1890  to  1895  Dr.  Linde,  Mr.  Tripler,  and  Dr.  Hampson  were  all 
working  along  much  the  same  line,  in  accordance  with  a  suggestion 
made  by  Sir  Wm.  Siemens  more  than  thirty  years  before;  namely, 
that  the  gas,  cooled  by  expansion  as  it  escapes  through  an  orifice, 
shall  cool  the  oncoming  gas  about  to  expand,  and  so  on,  thus  giving 
a  cumulative  effect.  Dr.  Linde,  however,  was  the  first  to  produce 
a  practical  machine. 

The  essential  parts  of  Linde's  apparatus  are  shown  in  Fig.  137. 
A  is  an  air  pump  which  takes  in  the  gas  (air,  e.g.)  through  valve 


VAPORIZATION  281 

E  at  about  16  atmospheres,  and  forces  it  under  a  pressure  of 
about  200  atmospheres  through  the  coiled  pipes  in  the  freezing 
bath  B.  From  B,  the  air  passes  successively  through  the  three 
concentric  pipes  or  tubes  F,  G,  and  H  in  the  vessel  C,  as  indicated 
by  the  arrows.  A  portion  of  the  air  from  G  returns  again  through 
pipe  I  and  valve  E  to  the  pump,  thus  completing  the  cycle.  The 
cycle  is  repeated  indefinitely  as  long  as  the  pump  is  operated.  It 
will  be  understood  that  the  freezing  bath  B  cools  the  air  which  has 
just  been  heated  by  compression.  It  also  "freezes  out"  most  of 
the  moisture  from  the  air.  The  pump  D  supplies  to  the  pump  A, 
under  a  pressure  of  16  atmospheres,  enough  air  to  compensate  for 
that  which  escapes  through  J  from  the  outer  tube  H,  and  also  for 
that  which  is  liquefied  and  collects  in  the  Dewar  flask  K. 

Explanation  of  the  Cooling  Action. — The  three  concentric  tubes 
F,  G,  and  H  (which  it  should  be  stated  are,  with  respect  to  the  rest 
of  the  apparatus,  very  much  smaller  than  shown,  and  in  practice 
are  coiled  in  a  spiral  within  C),  form  the  vital  part  of  the  appa- 
ratus. The  air,  as  it  passes  from  the  central  tube  F  through  valve 
L,  has  its  pressure  reduced  from  200  atmospheres  to  about  16 
atmospheres.  This  process  cools  it  considerably.  The  valves  are 
so  adjusted  that  about  4/5  of  this  cooled  air  flows  upward,  as 
indicated  by  the  curved  arrow,  through  G  (thereby  cooling  the 
downflowing  stream  in  F)  and  then  flows  through  7  back  to  the 
pump  A.  The  remaining  1/5  flows  directly  from  valve  L  through 
valve  M.  As  this  air  passes  through  valve  M  its  pressure  drops 
from  16  atmospheres  to  1  atmosphere,  producing  an  additional 
drop  in  temperature.  At  first  all  of  the  air  that  passes  through 
valve  M  passes  up  through  the  outer  tube  H  and  escapes  through 
J.  We  have  just  seen  that  the  downflowing  air  in  F  is  cooled  by 
the  upflowing  air  in  G,  and  as  this  downflowing  air  passes  through 
valve  L  it  is  still  further  cooled  (by  expansion),  and  therefore  as  it 
passes  up  through  G  it  still  further  cools  the  downflowing  stream  in 
F,  and  so  on.  Thus  both  streams  become  colder  and  colder  until 
so  low  a  temperature  is  reached  that  the  additional  cooling  pro- 
duced by  the  expansion  at  M  causes  part  (about  1/4)  of  the  air  that 
passes  through  M  to  liquefy  and  collect  in  the  Dewar  flask  K. 
From  K,  the  liquid  air  may  be  withdrawn  through  valve  N. 

Quite  recently  liquid  air  has  been  manufactured  at  the  rate  of 
about  one  quart  per  H.P.-hour  expended  in  operating  the  pumps. 

Properties  and  Effects  of  Liquid  Air. — Liquid  air  is  a  clear,  bluish 
liquid,  of  density  0.91  gm.  per  cm.3.  It  boils  at  a  temperature  of 


282  MECHANICS  AND  HEAT 

— 191°.4  C.  and  its  nitrogen  freezes  at  —210°,  its  oxygen  at 
—  227°.  It  is  attracted  by  a  magnet,  due  to  the  oxygen  which  it 
contains.  If  liquid  air  is  poured  into  water  it  floats  at  first;  but, 
due  to  the  fact  that  nitrogen  (density  0.85,  boiling  point  — 196°  C.) 
vaporizes  faster  than  oxygen  (density  1.13,  boiling  point  —183°), 
it  soon  sinks,  boiling  as  it  sinks,  and  rapidly  disappears.  Felt, 
if  saturated  with  liquid  air,  burns  readily. 

At  the  temperature  of  liquid  air,  mercury,  alcohol,  and  indeed 
most  liquids,  are  quickly  frozen.  Iron  and  rubber  become  almost 
as  brittle  as  glass;  while  lead  becomes  elastic,  i.e.,  more  like  steel. 

The  Dewar  Flask. — If  liquid  air  were  placed  in  a  closed  metal 
vessel  it  would  vaporize,  and  quickly  develop  an  enormous  pres- 
sure. Even  if  this  pressure  did  not  burst  the  container,  the  air 
would  soon  be  warmed  above  its  critical  temperature  and  cease  to 
be  a  liquid,  so  that  a  special  form  of  container  is  required.  Pro- 
fessor Dewar  performed  a  great  service  for  low-temperature 
research  when  he  devised  the  double-walled  flask  (K,  Fig.  137). 
In  such  a  container,  liquid  air  has  been  kept  for  hours  and  has 
been  shipped  to  a  considerable  distance.  The  space  between  the 
walls  is  a  nearly  perfect  vacuum,  which  prevents,  in  a  large  meas- 
ure, the  passage  of  heat  into  the  flask.  Silvering  the  walls  reflects 
heat  away  from  the  flask  and  therefore  improves  it.  These 
flasks  must  not  be  tightly  stoppered  even  for  an  instant  or  they 
will  explode,  due  to  the  pressure  caused  by  the  vaporization  of 
the  liquid  air.  The  constant  but  slow  evaporation  from  the  liquid 
air  keeps  it  cooled  well  below  its  critical  temperature,  in  fact  at 
about  — 191°  C.,  the  boiling  point  for  air  at  atmospheric  pressure. 

The  Thermal  Bottle— The  Thermal  Bottles  advertised  as  "Icy 
hot/'  etc.,  are  simply  Dewar  flasks  properly  mounted  to  prevent 
breakage.  They  will  keep  a  liquid  "warm  for  12  hours,"  or 
"cold  for  24  hours."  Observe  that  a  liquid  when  called  "warm" 
differs  more  from  room  temperature  than  when  called  "cold." 


CHAPTER  XVI 
TRANSFER  OF  HEAT 

207.  Three  Methods  of  Transferring  Heat.— Heat  may  be 
transferred  from  one  body  to  another  in  three  ways;  viz.,  by 
Convection,  by  Conduction,  and  by  Radiation. 

When  air  comes  in  contact  with  a  hot  stove  it  becomes  heated 
and  expands.  As  it  expands,  it  becomes  lighter  than  the  sur- 
rounding air  and  consequently  rises,  carrying  with  it  heat  to 
other  parts  of  the  room.  This  is  a  case  of  transfer  of  heat  by 
convection.  Obviously,  only  liquids  and  gases  can  transfer  heat 
by  convection. 

If  one  end  of  a  metal  rod  is  thrust  into  a  furnace,  the  other  end 
soon  becomes  heated  by  the  conduction  of  heat  by  the  metal  of 
which  the  rod  is  composed.  In  general,  metals  are  good  con- 
ductors, and  all  other  substances  relatively  poor  conductors, 
especially  liquids  and  gases. 

On  a  cold  day,  the  heat  from  a  bonfire  may  almost  blister  the 
face,  although  the  air  in  contact  with  the  face  is  quite  cool.  In 
this  case,  the  heat  is  transmitted  to  the  face  by  radiation.  The 
earth  receives  an  immense  amount  of  heat  from  the  sun,  although 
interplanetary  space  contains  no  material  substance  and  is  also 
very  cold.  This  heat  is  transmitted  by  radiation.  These  three 
methods  of  heat  transfer  will  be  taken  up  in  detail  in  subsequent 
sections. 

208.  Convection. — Heat  transfer  by  convection  is  utilized  in 
the  hot-air,  steam,  and  hot-water  systems  of  heating.     In  these 
systems  the  medium  of  heat  transfer  is  air,  steam,  and  water, 
respectively.     It  will  be  noted  in  every  case  of  heat  transfer  by 
convection,  that  the  heated  medium  moves  and  carries  the  heat  with 
it.     Thus,  in  the  hot-air  system,  an  air  jacket  surrounding  the 
furnace  is  provided  with  a  fresh-air  inlet  near  the  bottom;  while 
from  the  top,  air  pipes  lead  to  the  different  rooms  to  be  heated. 
As  the  air  between  the  jacket  and  the  furnace  is  heated  it  be- 
comes lighter  and  rises  with  considerable  velocity  through  the 
pipes  leading  to  the  rooms,  where  it  mingles  with  the  other  air 
of  the  room  and  thereby  warms  it. 

283 


284 


MECHANICS  AND  HEAT 


The  convection  currents  produced  by  a  hot  stove,  by  means  of 
which  all  parts  of  the  room  are  warmed,  are  indicated  by  arrows 
in  Fig.  138.  As  the  air  near  the  stove  becomes  heated,  and  there- 
fore less  dense,  it  rises,  and  the  nearby  air  which  comes  in  to  take 
its  place  is  in  turn  heated  and  rises.  As  the  heated  air  rises  and 
flows  toward  the  wall,  it  is  cooled  and  descends  as  shown. 

Fig.  139  illustrates  the  convection  currents  established  in  a 
vessel  of  water  by  a  piece  of  ice.  The  water  near  the  ice,  as  it  is 
cooled  becomes  more  dense  and  sinks.  Other  water  coming  in 
from  all  sides  is  in  turn  cooled  and  sinks,  as  indicated  by  the 
arrows. 

In  steam  heating,  pipes  lead  from  the  steam  boiler  to  the  steam 
radiators  in  the  rooms  to  be  heated.  Through  these  pipes,  the 


1 
vj 

\  / 

tM 

-^ 

\ 

«^ 

"    fS: 

FIG.  139. 

hot  steam  passes  to  the  radiators,  where  it  condenses  to  water. 
In  condensing,  the  steam  gives  up  its  heat  of  vaporization  and 
thereby  heats  the  radiator.  The  water  formed  by  the  condensa- 
tion of  the  steam  runs  back  to  the  boiler. 

In  the  hot-water  heating  system,  the  heated  water  from  the  boiler 
(B,  Fig.  140)  rises  through  pipes  leading  to  the  radiators  (C,  D, 
E,  and  F)  where  it  gives  up  heat,  thereby  warming  the  radiators, 
and  then  descends,  colder  and  therefore  denser,  through  other 
pipes  (G  and  H]  to  the  boiler,  where  it  is  again  heated.  This 
cycle  is  repeated  indefinitely.  The  current  of  water  up  one  pipe 
and  down  another  is  evidently  a  convection  current,  established 
and  maintained  by  the  difference  in  density  of  the  water  in  the 
two  pipes.  The  rate  of  flow  of  the  water  through  the  radiators, 
and  hence  the  heating  of  the  rooms,  may  be  controlled  by  the 
valves  c,  d,  e,  and/.  Hot  water  may  be  obtained  from  the  faucets 
/,  /,  K,  and  L.  The  tank  M  furnishes  the  necessary  pressure, 
allows  for  the  expansion  of  the  water  when  heated,  and  provides 


TRANSFER  OF  HEAT 


285 


a  safeguard  against  excessive  pressure  should  steam  form  in  the 
boiler. 

If  the  boiler  B  were  only  partly  filled  with  water,  steam  would 
pass  to  the  radiators  and  there  condense,  and  the  system  would 
become  a  steam-heating  system.  In  this  case  it  would  be  necessary 
to  provide  radiators  of  a  type  in  which  the  condensed  steam  would 
not  collect.  Usually  B  consists  of  water  tubes  surrounded  by 
the  flame. 

In  heating  a  vessel  of  water  by  placing  it  upon  a  hot  stove,  the 
water  becomes  heated  both  by  conduction  and  convection.  The 


FIG.  140. 


heat  passes  through  the  bottom  of  the  vessel  by  conduction  and 
heats  the  bottom  layer  of  water  by  conduction.  This  heated 
layer  is  less  dense  than  the  rest  of  the  water  and  rises  to  the 
surface,  carrying  with  it  a  large  quantity  of  heat.  Other  water, 
taking  its  place,  is  likewise  heated  and  rises  to  the  surface.  In 
this  way  convection  currents  are  established,  and  the  entire  body 
of  water  is  heated. 


286  MECHANICS  AND  HEAT 

Winds  are  simply  convection  currents  produced  in  the  air  by 
uneven  heating.  The  hotter  air  rises,  and  the  cooler  air  rushing 
in  to  take  its  place  is  in  turn  heated  and  rises.  This  inrush  of 
air  persists  so  long  as  the  temperature  difference  is  maintained, 
and  is  called  wind  (Chapter  XVII). 

209.  Conduction. — If  one  end  A  of  a  metal  rod  is  heated,  the 
other  end  B  is  supposed  to  become  heated  by  conduction  in  the 
following  manner.  The  violent  heat  vibrations  of  the  molecules 
at  the  end  A  cause  the  molecules  near  them  to  vibrate,  and  in 
like  manner  these  molecules,  after  having  begun  to  vibrate, 
cause  the  layer  of  molecules  adjacent  to  them  on  the  side  toward  B 
to  vibrate,  and  so  on,  until  the  molecules  at  the  end  B  are  vibrat- 
ing violently;  i.e.,  until  B  is  also  hot. 

This  vibratory  motion  is  readily  and  rapidly  transmitted  from 
layer  to  layer  of  the  molecules  of  metals;  therefore  metals  are 
said  to  be  good  conductors. 

Brick  and  wood  are  poor  conductors  of  heat,  which  fact  makes 
them  valuable  for  building  material.  Evidently  it  would  require 
a  great  deal  of  heat  to  keep  a  house  warm  if  its  walls  were  com- 
posed of  materials  having  high  heat  conductivity.  Asbestos 
is  a  very  poor  conductor  of  heat,  for  which  reason  it  is  much  used 
as  a  wrapping  for  steam  pipes  to  prevent  loss  of  heat,  and  also 
as  a  wrapping  for  hot  air  flues  to  protect  nearby  woodwork  from 
the  heat  which  might  otherwise  ignite  it. 

Clothing  made  of  wool  is  much  warmer  than  that  made  of 
cotton,  because  wool  is  a  much  poorer  conductor  of  heat  than 
cotton,  and  therefore  does  not  conduct  heat  away  from  the  body 
so  rapidly. 

Liquids,  except  mercury,  are  very  poor  conductors.  That 
water  is  a  poor  conductor  of  heat  may  be  demonstrated  by  the 
following  experiment.  A  gas  flame  is  directed  downward  against 
a  shallow  metal  dish  floating  in  a  vessel  of  water.  After  a  short 
time  the  water  in  contact  with  the  dish  will  boil,  while  the  water 
a  short  distance  below  experiences  practically  no  change  in  tem- 
perature, as  may  be  shown  by  thermometers  inserted.  It  will 
be  observed  that  convection  currents  are  not  established  when 
water  is  heated  from  above.  A  test  tube  containing  ice  cold 
water,  with  a  small  piece  of  ice  held  in  the  bottom,  may  be  heated 
near  the  top  until  the  top  layers  of  water  boil  without  apprecia- 
bly melting  the  ice. 

Gases  are  very  poor  conductors,  pf  heat — much  poorer  even  than. 


TRANSFER  OF  HEAT  287 

liquids.  The  fact  that  air  is  a  poor  conductor  is  frequently  made 
use  of  in  buildings  by  having  "dead  air"  spaces  in  the  walls. 
It  is  well  known  that  if  a  slight  air  space  is  left  between  the 
plaster  and  the  wall,  a  house  is  much  warmer  than  if  the  plaster 
is  applied  directly  to  the  wall.  If  a  brick  wall  is  wet  it  conducts 
heat  much  better  than  if  dry,  simply  because  its  pores  are  filled 
with  water  instead  of  with  air.  From  the  table  of  Thermal 
Conductivities  given  below,  it  will  be  seen  that  water  conducts 
heat  about  25  times  as  well  as  air.  Fabrics  of  a  loose  weave  are 
warmer  than  those  of  a  dense  weave  of  the  same  material  (except 
in  wind  protection),  because  of  the  more  abundant  air  space. 
A  wool-lined  canvass  coat  protects  against  both  wind  and  low 
temperature. 

Davy's  Safety  Lamp. — If  a  flame  is  directed  against  a  cold  metal 
surface,  it  will  be  found  that  the  metal  cools  below  the  combustion 
point  the  gases  of  which  the  flame  is  composed,  so  that  the  flame  does 


FIG.  141.  FIG.  141a. 

not  actually  touch  the  metal.  This  fact  may  be  demonstrated  by  pasting 
one  piece  of  paper  on  a  block  of  metal,  and  a  second  piece  on  a  block 
of  wood,  and  thrusting  both  into  a  flame.  The  second  piece  of  paper 
quickly  ignites,  the  first  does  not.  A  thin  paper  pail  quickly  ignites 
if  exposed  to  a  flame  when  empty,  but  not  when  filled  with  water. 

If  a  piece  of  wire  gauze  is  held  above  a  Bunsen  burner  or  other  gas 
jet,  the  flame  will  burn  above  the  gauze  only  (Fig.  141),  if  lighted  above, 
and  below  only  (Fig.  141a),  if  lighted  below.  The  flame  will  not  "strike 
through"  the  gauze  until  the  latter  reaches  red  heat.  Evidently,  the 
gas  (Fig.  141a)  as  it  passes  through  the  wire  gauze  is  cooled  below  its 
ignition  temperature.  If  a  lighted  match  is  now  applied  below  the  gauze 
(Fig.  141),  or  above  it  (Fig.  141a),  the  flame  burns  both  above  and  below 
as  though  the  gauze  were  absent. 

The  miner's  Safety  Lamp,  invented  by  Sir  Humphry  Davy,  has  its 
flame  completely  enclosed  by  iron  gauze.  The  explosive  fire-damp 
as  it  passes  through  the  gauze,  burns  within,  but  not  without,  and  thus 
gives  the  miner  warning  of  its  presence.  After  a  time  the  gauze  might 
become  heated  sufficiently  to  ignite  the  gas  and  cause  an  explosion. 

Boiler  "Scale." — The  incrustation  of  the  tubes  of  tubular 
boilers  with  lime,  etc.,  deposited  from  the  water,  is  one  of  the 


288  MECHANICS  AND  HEAT 

serious  problems  of  steam  engineering.  The  incrusted  material 
adds  to  the  thickness  of  the  walls  of  the  tubes,  and  is  also  a  very 
poor  conductor  of  heat  in  comparison  with  the  metal  of  the  tube. 
Consequently  it  interferes  with  the  transmission  of  the  heat  from 
the  heated  furnace  gases  to  the  water,  and  thereby  lowers  the 
efficiency  of  the  boiler.  Furthermore,  the  metal,  being  in  con- 
tact with  the  flame  on  the  one  side  and  the  "scale"  (instead  of  the 
water)  on  the  other,  becomes  hotter,  and  therefore  burns  out 
sooner  than  if  the  scale  were  prevented. 

210.  Thermal   Conductivity. — If   three   short  rods   of  similar 
size  and  length,  one  of  copper,  one  of  iron,  and  one  of  glass,  are 
held  by  one  end  in  the  hand  while  the  other 
^f^i^s.  f°  F°ld     end  is  thrust  into  the  gas  flame,  it  will  be  found 
that  the  copper  rod  quickly  becomes  unbear- 
ably hot,  the  iron  rod  less  quickly,  while  the 
glass  rod  does  not  become  uncomfortably  hot, 
however    long    it   is   held.     This    experiment 
shows  that  copper  is  a  better  conductor  than 
iron,  and  that  iron  is  a  better  conductor  than 
glass;  but  it  does  not  enable  us  to  tell  howmany 
FIG  142  times   better.     To   do  this  we  must  compare 

the  thermal  conductivities  of  the  two  metals, 
from  which  (see  table)  we  find  that  copper  conducts  about  five 
times  as  well  as  iron,  and  over  500  times  as  well  as  glass.  The 
fact  that  glass  is  such  a  very  poor  conductor  explains  why  the 
thin  glass  of  windows  is  so  great  a  protection  against  the  cold. 

If  one  face  of  a  slab  of  metal  (Fig.  142)  is  kept  at  a  higher 
temperature  than  the  other  face,  it  will  be  evident  that  the  num- 
ber of  calories  of  heat  Q  which  will  pass  through  the  slab  in 
T  seconds  will  vary  directly  as  the  time  T,  as  the  area  A  of  the 
face,  and  also  directly  as  the  difference  in  temperature  between 
the  two  faces  (i.e.,  ti—t2,  in  which  the  temperature  of  the  hotter 
face  is  t\°  and  the  colder,  £2°).  It  is  also  evident,  other  things 
being  equal,  that  less  heat  will  flow  through  a  thick  slab  than 
through  a  thin  one.  Indeed,  we  readily  see  that  the  quantity  Q 
will  vary  inversely  as  the  thickness  (d)  of  the  slab.  Accordingly 

we  have  Q  cc  A  T(t l  ~-^-  -  KA  T^j—  (85) 

in  which  K  is  a  constant,  whose  value  depends  upon  the  character 
of  the  material  of  which  the  slab  is  composed,  and  is  called  the 
Thermal  Conductivity  of  the  substance. 


TRANSFER  OF  HEAT 


289 


Since  Eq.  86  is  true  for  all  values  of  the  variables,  it  is  true  if 
we  let  A,  T,  (ti—tz),  and  d  all  be  unity.  This,  however,  would 
reduce  the  equation  to  K  =  Q.  Hence,  K  is  numerically  the  num- 
ber of  calories  of  heat  that  will  flow  in  unit  time  (the  second)  through 
a  slab  of  unit  area  and  unit  thickness  (i.e.,  through  a  cubic  centi- 
meter) if  its  two  opposite  faces  differ  in  temperature  by  unity  (1°C.). 

Temperature  Gradient. — Observe  that  — -5 —  is  the  fall  in  temperature 

per  centimeter  in  the  direction  of  heat  flow.  This  quantity  is  called 
the  Temperature  Gradient.  The  heat  conductivity,  then,  is  the  rate  of 
flow  of  heat  (calories  per  sec.)  through  a  conductor,  divided  by  the  product 
of  the  cross-sectional  area  and  the  temperature  gradient.  It  is  better  in 
determining  the  heat  conductivity  for  materials  which  are  good  con- 
ductors, such  as  the  metals,  to  use  a  rod  instead  of  a  slab. 

The  rod  is  conveniently  heated  at  one  end  by  steam  circulation,  and 
cooled  at  the  other  end  by  water  circulation.  The  temperature  of  the 
water  as  it  flows  past  the  end  of  the  rod  rises  from  £3°  to  £4°.  If  M 
grams  of  water  flows  past  in  T  seconds,  then  Q  =  M(tt  —  £3),  and  the 

Two  thermometers 

are  inserted  in  the  rod  at  a  distance  d  apart,  one  near  the  hot  end,  the 
other  near  the  cold  end.  Let  the  former  read  ti°  and  the  latter,  t2°. 

The  temperature  gradient  is,    then,  -^~.    The  remaining  quantity 

A  of  Eq.  86,  which  must  be  known  before  K  can  be  calculated,  is  the 
cross-sectional  area  of  the  rod.  If  the  rod  is  of  uniform  diameter  and 
is  packed  in  felt  throughout  its  length  to  prevent  loss  of  heat,  then  the 
rate  of  heat  flow,  and  also  the  temperature  gradient,  will  be  the  same  at 
all  points  in  the  rod. 

The  temperature  gradient  may  be  thought  of  as  forcing  heat  along  the 
rod,  somewhat  as  the  pressure  gradient  forces  water  along  a  pipe.  A 
few  thermal  conductivities  are  given  in  the  table  below. 

THERMAL  CONDUCTIVITIES 


rate  of  How  of  heat  through  the  rod  is  — — IT 


Substances 

Thermal 
conductivity 
K 

Substance 

Thermal 
conductivity 
K 

Silver 

1  006 

Marble 

0  0014 

Copper  
Aluminum  
Iron  .*  

0.88  to  0.96 
0.34 
0.16  to  0.20 
0  016 

Water  
Hydrogen.... 
Paraffine.  .  .  . 
Air 

0.0014 
0.00033 
0.00025 
0  000056 

Glass..  . 

0.0015 

Flannel.  .  . 

0.000035 

The  value  of  the  thermal  conductivity  varies  greatly  in  some 
cases  for  different  specimens  of  the  same  substance.     Thus,  for 

19 


290  MECHANICS  AND  HEAT 

hard  steel,  it  is  about  one-half  as  large  as  for  soft  steel,  and 
about  one-third  as  large  as  for  hard  steel.  Different  kinds  of 
copper  give  different  results.  The  values  given  in  the  table  are 
approximate  average  values. 

211.  Wave  Motion. — The  kinds  of  wave  motion  most  com- 
monly met  are  three  in  number,  typified  by  water  waves,  sound 
waves,  and  ether  waves.  The  beautiful  waves  which  travel  over 
a  field  of  grain  on  a  windy  day,  are  quite  similar  to  water  waves  in 
appearance,  and  similar  to  all  waves  in  one  respect;  namely, 
that  the  medium  (here  the  swaying  heads  of  grain)  does  not  move 
forward,  but  its  parts,  or  particles  simply  oscillate  to  and  fro 
about  their  respective  equilibrium  positions. 

Water  Waves. — There  are  many  kinds  of  water  waves;  varying 
in  form  from  the  smooth  ocean  "swell"  due  to  a  distant  storm,  to 
the  "choppy"  storm-lashed  billows  of  the  tempest;  and  varying 
in  size  from  the  large  ocean  waves  20  ft.  or  more  in  height,  to  the 
tiny  ripples  that  speed  over  a  still  pond  before  a  sudden  gust  of 
wind.  The  Tide  (Sec.  30)  consists  of  two  wave  crests  on  opposite 
sides  of  the  earth,  which  travel  around  the  earth  in  about  25  hrs. 
Consequently,  at  the  equator,  the  wave  length  is  over  12,000  miles, 
and  the  velocity  about  1000  miles  per  hour. 

Restoring  Force. — In  all  cases  of  wave  motion,  at  least  in 
material  media,  there  must  be  a  restoring  force  developed  which 
acts  upon  the  displaced  particle  of  the  medium  in  such  a  direction 
as  to  tend  to  bring  it  back  to  its  equilibrium  position.  As  the 
head  of  grain  sways  to  and  fro,  the  supporting  stem,  alternately 
bent  this  way  and  that,  furnishes  the  restoring  force.  As  the 
vibrating  particle  reaches  its  equilibrium  position,  it  has  kinetic 
energy  which  carries  it  to  the  position  of  maximum  displacement 
in  the  opposite  direction.  Thus  the  swaying  head  of  grain 
when  the  stem  is  erect  is  in  equilibrium,  but  its  velocity  is  then  a 
maximum  and  it  moves  on  and  again  bends  the  stem. 

In  the  case  of  large  water  waves,  the  restoring  force  is  the  gravi- 
tational pull  which  acts  downward  on  the  "crest,"  and  the  buoy- 
ant force  which  acts  upward  on  the  "trough"  of  the  wave.  These 
waves  are  often  called  gravitational  water  waves.  In  the  case 
of  fine  ripples,  the  restoring  force  is  mainly  due  to  surface  tension. 
The  velocity  of  long  water  waves  increases  with  the  wave  length 
(distance  from  crest  to  crest),  while  with  ripples,  the  reverse  is 
true;  i.e.,  the  finer  the  ripples  are,  the  faster  they  travel. 

Sound  Waves. — As  the  prong  of  a  tuning  fork  vibrates  to  and 


TRANSFER  OF  HEAT  291 

fro,  its  motion  in  one  direction  condenses  the  air  ahead  of  it; 
while  its  return  motion  rarefies  the  air  at  the  same  point.  These 
condensations  and  rarefactions  travel  in  all  directions  from  the 
fork  with  a  velocity  of  about  1100  ft.  per  sec.,  and  are  called 
Sound  Waves.  Obviously,  if  the  tuning  fork  vibrated  1100  times 
per  sec.,  one  condensation  would  be  one  foot  from  the  tuning 
fork  when  the  next  condensation  started;  while  if  the  fork 
vibrated  110  times  per  sec.  this  distance  between  Condensations, 
called  the  wave  length  X,  would  be  10  ft.  In  other  words  the 
relation  v  =  n\  is  true,  in  which  v  is  the  velocity  of  sound,  and  n, 
the  number  of  vibrations  of  the  tuning  fork  per  second.  Sound 
waves  are  given  off  by  a  vibrating  body,  and  are  transmitted  by 
any  elastic  medium,  such  as  air,  water,  wood,  and  the  metals. 
The  velocity  varies  greatly  with  the  medium,  but  the  relation 
v  =  n\  always  holds. 

Ether  Waves. — Ether  waves  consist  in  vibrations  of  the  Ether 
(Sec.  214),  a  medium  which  is  supposed  to  pervade  all  space  and 
permeate  all  materials.  These  vibrations  are  produced,  in  the 
case  of  heat  or  light  waves,  by  atomic  vibrations  in  a  manner  not 
understood.  The  ether  waves  used  in  wireless  telegraphy  are 
produced  by  special  electrical  apparatus  which  we  cannot  discuss 
here. 

Ether  waves  are  usually  grouped  in  the  following  manner. 
Those  which  affect  the  eye  (i.e.,  produce  the  sensation  of  light) 
are  called  light  waves,  while  those  too  long  to  affect  the  eye  are 
called  heat  waves.  Those  waves  which  are  too  short  to  affect  the 
eye  do  affect  a  photographic  plate,  and  are  sometimes  called 
actinic  waves.  It  should  not  be  inferred  that  light  waves  do  not 
produce  heat  or  chemical  (e.g.,  photographic)  effects,  for  they  do 
produce  both.  Certain  waves  which  are  still  longer  than  heat 
waves,  and  which  are  produced  electrically,  are  called  Hertz 
waves.  These  waves  are  the  waves  employed  in  wireless  teleg- 
raphy. They  were  discovered  in  1888  by  the  German  physicist, 
H.  R.  Hertz  (1857-94). 

The  longest  ether  waves  that  affect  the  eye  are  those  of  red 
light  (X  =  1/35000  in.  approx.).  Next  in  order  of  wave  length  are 
orange,  yellow,  green,  blue,  and  violet  light.  The  wave  length 
of  violet  light  is  about  one-half  that  of  red,  while  ultra-violet 
light  of  wave  length  less  than  one-third  that  of  violet  has  been 
studied  by  photographic  means.  An  occupant  of  a  room  flooded 
with  ultra-violet  light  would  be  in  total  darkness,  and  yet  with  a, 


292  MECHANICS  AND  HEAT 

camera,  using  a  short  exposure,  he  could  take  a  photograph  of  the 
objects  in  the  room.  The  wave  lengths  longer  than  those  of  red 
light,  up  to  about  1/500  inch,  have  been  much  studied,  and  are 
called  heat  waves,  or  infra-red.  It  is  interesting  to  note  that  the 
shortest  Hertz  waves  that  have  been  produced  are  but  little  longer 
than  the  longest  heat  waves  that  have  been  studied.  If  this 
small  "gap"  were  filled,  then  ether  waves  varying  in  length  from 
several  miles  to  1/200000  in.  would  be  known. 

Since  the  velocity  v  for  all  ether  waves  is  186,000  miles  per  sec., 
the  frequency  of  vibration  n  for  any  given  wave  length  X  is  quickly 
found  from  the  relation  v  =  n\.  Thus  the  frequency  of  vibration 
of  violet  light  for  which  X  =  1/70000  in.  is  about  800,000,000,- 
000,000.  This  means  that  the  source  of  such  light,  the  vibrating 
atom,  or  atomic  particle  (electron)  sends  out  800,000,000,000,000 
vibrations  per  second! 

Direction  of  Vibration. — A  water  wave  in  traveling  south,  let 
us  say,  would  appear  to  cause  the  water  particles  to  vibrate  up 
and  down.  Careful  examination,  however,  will  show  that  there  is 
combined  with  this  up-and-down  motion  a  north-and-south 
motion;  so  that  any  particular  particle  is  seen  to  describe  approxi- 
mately a  circular  path.  A  sound  wave  traveling  south  causes 
the  air  particles  to  vibrate  to  and  fro  north  and  south;  while  an 
ether  wave  traveling  south  would  cause  the  ether  particles  to 
vibrate  up  and  down  or  east  and  west,  or  in  some  direction  in  a 
plane  which  is  at  right  angles  to  the  direction  in  which  the  wave  is 
traveling.  For  this  reason,  the  ether  wave  is  said  to  be  a  Trans- 
verse Wave  and  the  sound  wave,  a  Longitudinal  Wave.  The 
phenomena  of  polarized  light  seem  to  prove  beyond  question 
that  light  is  a  transverse  wave. 

212.  Interference  of  Wave  Trains. — A  succession  of  waves,  following 
each  other  at  equal  intervals,  constitutes  a  wave  train.  A  vibrating 
tuning  fork  or  violin  string,  or  any  other  body  which  vibrates  at  a 
constant  frequency,  gives  rise  to  a  train  of  sound  waves.  Two  such 
wave  trains  of  different  frequency  produce  interference  effects,  known 
as  beats,  which  are  familiar  to  all. 

Interference  of  Sound  Waves. — Let  a  tuning  fork  A  of  200  vibrations 
per  second  be  sounded.  The  train  of  waves  from  this  fork,  impinging 
upon  the  ear  of  a  nearby  listener,  will  cause  the  tympanum  of  his  ear 
to  be  alternately  pushed  in  and  out  200  times  per  sec.,  thus  giving 
rise  to  the  perception  of  a  musical  tone  of  uniform  intensity.  If,  now, 
a  second  fork  B  of,  say  201  vibrations  per  second,  is  sounded,  the  train 


TRANSFER  OF  HEAT  293 

of  waves  from  it,  let  us  say  the  "  B  train,"  will  interfere  with  the  "A 
train"  and  produce  an  alternate  waxing  and  waning  in  the  intensity 
of  the  sound,  known  as  "beats."  In  this  case  there  would  be  1  beat 
per  sec.  For,  consider  an  instant  when  a  compressional  wave  from  the 
A  train  and  one  from  the  B  train  both  strike  the  tympanum  together. 
This  will  cause  the  tympanum  to  vibrate  through  a  relatively  large  dis- 
tance, i.e.,  it  will  cause  it  to  have  a  vibration  of  large  amplitude,  and  a 
loud  note  (maximum)  will  be  heard.  (The  amplitude  of  a  vibration  is 
half  the  distance  through  which  the  vibrating  body  or  particle,  as  the 
case  may  be,  moves  when  vibrating;  in  other  words,  it  is  the  maximum 
displacement  of  the  particle  from  its  equilibrium  position.)  One-half 
second  later,  a  compressional  wave  from  the  A  train  and  a  rarefaction  from 
the  B  train  will  both  strike  the  tympanum.  Evidently  these  two  dis- 
turbances, which  are  said  to  be  out  of  phase  by  a  half  period,  will  produce 
but  little  effect  upon  the  tympanum,  in  fact  none  if  the  two  wave  trains 
have  exactly  equal  amplitudes.  Consequently,  a  minimum  in  the  tone 
is  heard.  Still  later,  by  1/2  sec.,  the  two  trains  reach  the  ear  exactly  in 
phase,  and  another  maximum  of  intensity  in  the  tone  is  noted,  and  so  on. 
Obviously,  for  a  few  waves  before  and  after  the  maximum,  the  two  trains 
of  waves  will  be  nearly  in  phase,  and  a  fairly  loud  tone  will  be  heard. 
This  tone  dies  down  gradually  as  the  waves  of  the  two  trains  get  more 
and  more  out  of  phase  with  each  other,  until  the  minimum  is  reached. 

Had  the  tuning  forks  differed  by  10  vibrations  per  second,  there  would 
have  been  10  beats  per  second.  To  tune  a  violin  string  to  unison  with 
a  piano,  gradually  increase  (or  decrease)  the  tension  upon  it  until  the 
beats,  which  come  at  longer  and  longer  intervals,  finally  disappear 
entirely.  If  increasing  the  tension  produces  more  beats  per  second, 
the  string  is  already  of  too  high  pitch. 

Interference  of  Light  Waves. — By  a  proper  arrangement,  two  trains  of 
light  waves  of  equal  frequency  and  equal  amplitude  maybe  produced. 
If  these  two  trains  fall  upon  a  photographic  plate  from  slightly  different 
directions,  they  will  reinforce  each  other  at  some  points  of  the  film, 
and  annul  each  other  at  other  points.  For  certain  portions  of  the  plate, 
the  two  trains  are  constantly  one-half  period  out  of  phase.  Such 
portions  are  in  total  darkness,  and  therefore  remain  clear  when  the 
plate  is  "developed,"  producing,  with  the  alternate  "exposed"  strips,  a 
beautiful  effect.  We  here  have  the  strange  anomaly  of  light  added  to 
light  producing  darkness,  for  either  beam  alone  would  have  affected 
the  entire  photographic  plate. 

213.  Reflection  and  Refraction  of  Waves.— In  Fig.  143,  let 
AB  be  a  stone  pier,  and  let  abc,  etc.,  be  water  waves  traveling  in 
the  direction  bO.  Then  a'b'c',  etc.  (dotted  lines),  will  be  the 
reflected  water  waves,  and  will  travel  in  the  direction  Ob',  such 
that  bO  and  Ob'  make  equal  angles  0i  and  02  with  the  normal 


294 


MECHANICS  AND  HEAT 


(NO)  to  the  pier.     This  important  law  of  reflection  is  stated  thus: 
The  angle  of  reflection  (02)  is  equal  to  the  angle  of  incidence  (0i). 

If  AB  is  a  mirror  and  abc,  etc.,  light  waves,  or  heat  waves, 
then  the  construction  will  show  accurately  the  reflection  of  light 
or  heat  waves,  as  the  case  may  be. 

Proof:  If  the  reflected  wave  has  the  same  velocity  as  the  inci- 
dent wave,  which  is  strictly  true  in  the  case  of  heat  and  light, 
then,  while  the  incident  light  (let  us  say)  travels  from  a\  to  a2, 
the  reflected  light  will  travel  from  c\  to  c2.  The  triangles 
and  c^ttzCi  will  be  not  only  similar,  but  equal.  Therefore  63 
But  63  =  61  and  04  =  02,  hence  0i  =  02,  which  was  to  be  proved. 


Refraction. — Let  abc  (Fig.  144)  represent  a  light  wave  or  a 
heat  wave,  traveling  in  the  direction  60.  Then,  as  the  portion  a 
reaches  a',  portion  c  will  have  reached  c'  instead  of  c".  The  ratio 
cc'/cc"  is  about  3/4,  since  light  and  heat  radiation  travel  about 
3/4  as  fast  in  water  as  in  air.  The  reciprocal  of  this  ratio,  i.e., 
the  velocity  in  air  divided  by  the  velocity  in  water,  is  called  the 
index  of  refraction  for  water.  The  index  of  refraction  for  glass 
varies  with  the  kind  of  glass  and  the  length  of  the  wave,  from 
about  1.5  to  2.  Since  the  ray  is  always  normal  to  the  wave 
front,  the  ray  Ob'  deviates  from  the  direction  60  by  the  angle 
a,  called  the  angle  of  deviation.  The  fact  that  the  ray  bends 
sharply  downward  as  it  enters  the  water,  accounts  for  the  apparent 
sharp  upward  bending  of  a  straight  stick  held  in  a  slanting 
position  partly  beneath  the  surface  of  the  water. 


TRANSFER  OF  HEAT  295 

The  fact  that  light  and  heat  radiation  travel  more  slowly  in 
glass  than  in  air,  thus  causing  all  rays  which  strike  the  glass 
obliquely  to  be  deviated,  makes  possible  the  focusing  of  a  bundle 
of  rays  at  a  point  by  means  of  a  glass  lens,  and  therefore  makes 
possible  the  formation  of  images  by  lenses.  Since  practically  all 
optical  instruments  consist  essentially  of  a  combination  of  lenses, 
we  see  the  great  importance  of  the  refractive  power  of  glass  and 
other  transparent  substances.  Indeed  were  it  not  for  the  fact 
that  light  travels  more  slowly  through  the  crystalline  lens  of  the 
eye  than  through  air,  vision  itself  would  be  ii 


FIG.  144. 

The  production  of  the  rainbow  and  prismatic  colors  in  general 
depends  upon  the  fact  that  the  velocity  of  light  in  glass,  water, 
etc.,  depends  upon  the  wave  length,  being  greatest  for  red  and 
least  for  violet.  Consequently  red  light  is  deviated  the  least,  the 
violet  the  most. 

214.  Radiation. — If  a  glowing  incandescent  lamp  is  placed 
under  the  receiver  of  an  air  pump,  it  will  be  found  that  it  gives  off 
heat  and  heats  the  receiver,  whether  the  receiver  contains  air  or 
a  vacuum.  It  is  evident,  then,  that  the  air  is  not  the  medium 
of  transfer  of  heat  by  radiation.  Likewise,  in  the  case  of  heat 
and  light  received  from  the  sun,  the  medium  of  transfer  cannot  be 
air.  Since  the  transmission  of  a  vibratory  motion  from  one  point 
to  another  requires  an  intervening  medium,  physicists  have  been 
led  to  postulate  the  Ether  as  such  a  medium,  and  have  ascribed  to 
it  such  properties  as  seem  best  to  explain  the  observed  phenomena. 
The  ether  is  supposed  to  fill  all  space  and  also  to  permeate  all 


296  MECHANICS  AND  HEAT 

materials.  Thus  we  know  that  the  heat  of  the  sun  passes  readily 
through  glass  by  radiation.  This  is  effected,  however,  by  the 
ether  in  the  glass  and  not  by  the  glass  itself.  Indeed  the  glass 
molecules  prevent  the  ether  from  transmitting  the  radiation  so 
well  as  it  would  if  the  glass  were  absent. 

While  immense  quantities  of  heat  are  transferred  from  the  sun 
to  the  earth  by  radiation,  it  is  well  to  call  attention  to  the  fact 
that  what  we  call  radiant  heat  or  heat  radiation,  is  not  strictly  heat, 
but  energy  of  wave  motion.  Radiant  heat  does  not  heat  the 
medium  through  which  it  passes  (unless  it  is  in  part  absorbed), 
but  heats  any  body  which  it  strikes — a  good  reflector  least,  a  lamp- 
black surface  most.  Both  heat  radiation  and  light  may  be  re- 
flected, and  also  refracted  (Sec.  213).  The  moon  and  the  planets, 
in  the  main,  shine  by  reflected  sunlight.  We  see  all  objects 
which  are  not  self-luminous,  by  means  of  irregularly  (scattering) 
reflected  light.  At  South  Pasadena,  Cal.,  a  10-H.P.  steam  engine 
is  run  by  a  boiler  which  is  heated  by  means  of  sunlight  reflected 
from  a  great  number  of  properly  placed  mirrors. 

215.  Factors  in  Heat  Radiation. — It  has  been  shown  experi- 
mentally that  the  higher  the  temperature  of  a  body  becomes,  the 
faster  it  radiates  heat  energy.  Obviously,  the  amount  of  heat 
radiated  in  a  given  time  will  also  be  proportional  to  the  amount 
of  heated  surface.  It  has  also  been  found  that  two  metal  spheres, 
A  and  B,  alike  as  to  material,  size  and  weight,  but  differing  in 
finish  of  surface,  have  quite  different  radiating  powers.  Thus  if 
A  is  highly  polished,  so  as  to  have  a  mirror-like  surface,  while  B 
is  coated  with  lamp  black,  it  will  be  found  that  B  radiates  heat 
much  faster  than  A.  This  is  easily  tested  by  simply  heating  A 
and  B  to  the  same  temperature  and  then  suspending  them  to  cool. 
It  will  be  found  that  B  cools  much  more  rapidly  than  A,  which 
shows  that  B  parts  with  its  heat  more  quickly,  i.e.,  radiates  better, 
than  A.  A  lamp-black  surface  is  about  the  best  radiating  surface, 
while  a  polished  mirror  surface  is  about  the  poorest.  The  radiat- 
ing powers  of  other  substances  lie  between  those  of  these  two. 
From  the  above  discussion,  we  see  that  the  high  polish  of  the 
nickel  trimmings  of  stoves  decreases  their  efficiency  somewhat. 

Prevost's  Theory  of  Heat  Exchanges. — According  to  this  theory, 
a  body  radiates  heat  to  surrounding  bodies  whether  it  is  warmer 
than  they  or  colder.  In  the  former  case  it  radiates  more  heat  to 
the  surrounding  bodies  than  it  receives  from  them,  and  its  tempera- 
ture falls;  while  in  the  latter  case  it  radiates  less  heat  than  it 


TRANSFER  OF  HEAT  297 

receives,  and  its  temperature  rises.  The  fall  in  temperature 
experienced  by  a  body  when  placed  near  ice,  a  result  which  would 
at  first  seem  to  indicate  that  cold  can  be  radiated  and  that  it  is 
not  therefore  merely  the  absence  of  heat,  is  easily  explained  by 
this  exchange  theory.  The  body  radiates  heat  no  faster  to  the  ice 
than  it  would  to  a  warmer  body,  but  it  receives  less  in  return,  and 
therefore  becomes  colder. 

Laws  of  Cooling.  —  Newton  considered  that  the  amount  of  heat 
H,  radiated  from  a  body  of  temperature  t,  to  its  surroundings  of 
temperature  t',  was  proportional  to  the  difference  in  temperature; 
i.e., 

H  =  K(t-t') 

in  which  K  is  a  constant,  depending  upon  the  size  and  character 
of  the  surface.  This  law  is  very  nearly  true  for  slight  differences 
in  temperature  only.  Thus  a  body  loses  heat  almost  exactly 
twice  as  fast  when  2°  warmer  than  its  surroundings  as  it  does  when 
1°  warmer.  Experiment,  however,  shows  that  if  this  tempera- 
ture difference  is,  say,  20°,  the  amount  of  heat  radiated  is  more 
than  20  times  as  great  as  when  it  is  1°. 

The  quite  different  law,  expressed  by  the  equation 


is  due  to  Stefan,  and  is  known  as  Stefan's  Law.  In  this  equation 
T  and  T'  are  the  temperatures  of  the  body  and  its  surroundings, 
respectively,  on  the  absolute  scale.  Stefan's  law,  applied  to 
radiation  by  black  bodies,  accords  with  experimental  results. 

216.  Radiation  and  Absorption.  —  It  has  been  found  experi- 
mentally that  surfaces  which  radiate  heat  rapidly  when  hot, 
absorb  heat  rapidly  when  cold.  Thus  if  the  two  metal  spheres 
mentioned  in  Sec.  215  were  placed  in  the  sunshine,  B  would  be 
warmed  very  much  more  quickly  than  A.  Evidently  the  same 
amount  of  solar  heat  radiation  would  strike  each,  but  A  reflects 
more  and  consequently  absorbs  less  than  B,  which  has  smaller 
reflecting  power.  There  is  a  close  proportionality  between  radia- 
tion and  absorption.  For  example,  if  B,  when  hot,  loses  heat  by 
radiation  twice  as  fast  as  A  does  when  equally  hot,  then  if  both  are 
equally  cold  and  are  placed  in  the  sunshine,  B  will  absorb  heat 
practically  twice  as  fast  as  A.  That  is,  good  absorbers  of  heat 
(when  cold)  are  good  radiators  of  heat  (when  hot).  If  two 
thermometers,  one  of  which  has  its  bulb  smoked  until  black, 


298  MECHANICS  AND  HEAT 

are  placed  side  by  side  in  the  sunshine,  the  one  with  the  blackened 
bulb  will  indicate  a  higher  temperature  than  the  other. 

217.  Measurement  of   Heat   Radiation. — By    means  of   the 
thermopile  (Sec.  174),  and  other  sensitive  devices,  such  as  the 
bolometer,  many  measurements  of  intensity  of  heat  radiation 
have  been  made.     When  white  light,  e.g.,  sunlight,  passes  through 
a  prism,  the  different  colors  of  light  take  slightly  different  direc- 
tions, and  a  "spectrum"  of  the  colors,  red,  orange,  yellow,  green, 
blue,  and  violet,  is  produced. 

By  exposing  the  bolometer  successively  in  the  violet,  blue, 
green,  yellow,  orange  and  the  red,  and  then  moving  it  still  farther, 
into  the  invisible  or  infra-red  part  of  the  spectrum,  it  is  found  that 
the  radiant  energy  increases  with  the  wave  length,  and  reaches  a 
maximum  in  the  infra-red.  In  other  words,  the  wave  length  of 
the  sun's  radiation  which  contains  the  most  energy  is  slightly 
greater  than  that  of  the  extreme  red.  It  has  been  found  by  experi- 
ment, using  various  sources  of  known  temperature  for  producing 
the  light,  that  the  wave  length  of  maximum  energy  is  shorter,  the 
hotter  the  source.  From  these  considerations  the  temperature  of 
the  sun  is  estimated  to  be  about  6000°  C.  In  such  experiments,  a 
rock-salt  prism  must  be  used,  since  glass  absorbs  infra-red  radia- 
tions to  a  great  extent. 

218.  Transmission  of  Heat  Radiation  Through  Glass,  Etc.— 
Just  as  light  passes  readily  through  glass  and  other  transparent 
substances,  so  heat  radiation  passes  readily  through  certain  sub- 
stances.    In  general,  substances  transparent  to  light  are  also  trans- 
parent to  heat  radiation,  but  there  are  some  exceptions  to  this  rule. 

A  thin  pane  of  glass  gives  very  little  protection  from  the  sun's 
heat,  but  if  held  between  the  face  and  a  hot  stove  it  is  a  great 
protection.  It  may  be  remarked  that  in  the  former  case,  the 
glass  is  not  noticeably  warmed,  while  in  the  latter  case  it  is 
warmed.  It  is  apparent,  then,  that  the  glass  transmits  solar 
radiation  better  than  it  does  the  radiation  from  the  hot  stove. 
This  selective  transmission  of  radiation  is  really  due  to  "selective 
absorption."  The  glass  absorbs  a  greater  percentage  of  the  ra- 
diation in  the  latter  case  than  in  the  former,  which  accounts  not 
only  for  the  fact  that  it  transmits  less  heat  in  the  case  of  the  radia- 
tion from  the  stove,  but  also  for  the  fact  that  it  is  heated  more. 

In  this  connection,  we  may  state  that  it  has  been  shown  by 
experiment  that  a  body,  say  a  piece  of  iron,  when  heated  to  a  white 
heat,  gives  off  simultaneously  heat  waves  varying  greatly  in 


TRANSFER  OF  HEAT  299 


length.  As  it  is  heated  more  and  more,  it  gives  off  more  and  more 
energy  of  all  wave  lengths;  but  the  energy  of  the  shorter  wave 
lengths  increases  most  rapidly.  Accordingly,  the  wave  length  of 
maximum  energy  becomes  shorter  the  hotter  the  source,  as  stated 
in  Sec.  217. 

Just  before  the  iron  reaches  "red  heat"  the  heat  waves  are  all 
too  long  to  be  visible.  As  it  becomes  hotter,  somewhat  shorter 
waves,  corresponding  to  red  light,  are  given  off,  and  we  say  that 
the  iron  is  "red  hot."  If  heated  to  a  still  higher  temperature,  so 
as  to  give  off  a  great  deal  of  light,  in  fact  light  of  all  different  wave 
lengths,  we  say  that  it  is  "white  hot."  A  hot  stove,  then,  gives 
off,  in  the  main,  very  long  heat  waves;  while  the  sun,  which  is 
intensely  heated,  gives  off  a  great  deal  of  its  heat  energy  in  the 
short  wave  lengths. 

The  above-mentioned  fact,  that  glass  affords  protection  from 
the  heat  radiation  from  a  stove,  and  no  appreciable  protection  in 
the  case  of  solar  radiation,  is  explained  by  saying  that  glass  trans- 
mits short  heat  waves  much  better  than  long  heat  waves,  i.e., 
glass  is  more  transparent  to  short  than  to  long  heat  waves.  More 
strictly,  it  might  be  said  that  glass  does  not  prevent  the  trans- 
mission of  short  heat  waves  by  the  ether  permeating  it,  to  so  great 
an  extent  as  it  does  the  long  heat  waves. 

The  "Hotbed. " — The  rise  in  temperature  of  the  soil  in  a  Hotbed, 
when  the  glass  cover  is  on,  above  what  it  would  be  if  the  glass 
were  removed,  is  in  part  due  to  this  behavior  of  glass  in  the  trans- 
mission of  heat  radiation.  The  greater  part  of  the  solar  heat  that 
strikes  the  glass,  being  of  short  wave  length,  passes  readily 
through  the  glass  to  the  soil,  which  is  thereby  warmed.  As  the 
soil  is  warmed,  it  radiates  heat  energy,  but  in  the  form  of  long 
heat  waves  which  do  not  readily  pass  through  the  glass,  and  hence 
the  heat  is  largely  retained.  The  fact  that  the  glass  prevents  a 
continual  stream  of  cold  air  from  flowing  over  the  soil  beneath  it, 
and  still  permits  the  sun  to  shine  upon  the  soil,  accounts  in  large 
part  for  its  effectiveness. 

"Smudging"  of  Orchards. — Very  soon  after  sunset,  blades  of 
grass  and  other  objects,  through  loss  of  heat  by  radiation,  usually 
become  cool  enough  to  precipitate  part  of  the  moisture  of  the  air 
upon  them  in  the  form  of  Dew  (Sees.  220,  221).  It  is  well  known 
that  heavy  dews  form  when  the  sky  is  clear.  If  the  sky  is  over- 
cast, even  by  fleecy  clouds,  a  portion  of  the  radiated  heat  is 
reflected  by  the  clouds  back  to  the  earth,  and  the  cooling  of 


300  MECHANICS  AND  HEAT 

objects,  and  consequently  the  formation  of  dew  upon  them,  is 
less  marked. 

Many  fruit  growers  have  placed  in  the  orchard,  a  thermostat, 
so  adjusted  that  an  alarm  is  sounded  when  the  freezing  point  is 
approached.  As  soon  as  the  alarm  is  sounded,  the  "smudge" 
fires  (coal,  coal  oil,  etc.)  are  started.  These  fires  produce  a  thin 
veil  of  smoke,  which  hovers  over  the  orchard  and  protects  it  from 
frost,  somewhat  as  a  cloud  would.  In  addition  to  the  protection 
afforded  by  the  smoke,  the  considerable  amount  of  heat  developed 
by  the  fires  is  also  important.  If  the  wind  blows,  such  protec- 
tion is  much  less  effective.  Frosts,  however,  usually  occur  during 
still,  clear  nights. 

219.  The  General  Case  of  Heat  Radiation  Striking  a  Body.— 
Heat  radiation,  e.g.,  solar  radiation,  when  it  strikes  a  body,  is 
in  general  divided  into  three  parts:  the  part  (a)  which  is  reflected; 
the  part  (6)  which  is  absorbed  and  therefore  tends  to  heat  the  body; 
and  the  part  (c)  which  is  transmitted,  or  passes  through  the  body. 
The  sum  of  these,  i.e.,  a+6-f-c,  is  of  course  equal  to  the  original 
energy  that  strikes  the  body.  In  some  cases,  the  part  reflected 
is  large,  e.g.,  if  the  body  has  polished  surfaces.  In  other  cases, 
the  part  absorbed  is  large,  e.g.,  for  lamp  black  or,  in  general,  for 
dull  surfaces,  and  also  for  certain  partially  transparent  substances. 
The  part  transmitted  is  large  for  quartz  and  rock  salt;  much 
smaller  for  glass,  water  and  ice,  and  absent  for  metals  unless 
they  are  in  the  form  of  exceedingly  thin  foil. 


PROBLEMS 

1.  If  a  piece  of  plate  glass  80  cm.  in  length,  50  cm.  in  width,  and  1.2  cm. 
in  thickness,  is  kept  20°  C.  hotter  on  one  side  than  on  the  other,  how  many 
calories  of  heat  pass  through  it  every  minute  by  conduction  alone? 

2.  A  copper  vessel,  the  bottom  of  which  is  0.2  cm.  thick,  has  an  area  of 
400  cm.2,  and  contains  3  kilograms  of  water.     What  will  be  the  temperature 
rise  of  the  water  in  it  in  1  minute,  if  the  lower  side  of  the  bottom  is  kept 
3°  C.  warmer  than  the  upper  side  of  the  bottom? 

3.  Assuming  the   sun  to  be  directly  over  head,  what  power  (in  H.P.) 
does  it  radiate  in  the  form  of  heat  upon  an  acre  of  land  at  noon.     See  Sec. 
161. 

4.  A  wall  10  in.  thick  is  made  of  a  material,  the  thermal  conductivity  of 
which  is  0.001 12.     The  wall  is  made  "twice  as  warm"  by  rebuilding  it  with 
an  additional  thickness  of  "dead  air"  space.     Find  the  thickness  of  the  air 
space.     (In  practice,  convection  currents  diminish  considerably  the  effective- 
ness of  so-called  "dead  air"  spaces.) 


TRANSFER  OF  HEAT  301 

5.  How  many  pounds  of  steam  at  140°  C.  (heat  of  vap.  509  cal.  per  gm.) 
will  a  boiler  furnish  per  hour  if  it  has  1000  sq.  ft.  of  heating  surface  of  iron 
(thermal  conductivity  0.16)  0.25  in.  in  thickness,  which  is  kept  5°  hotter 
next  the  flame  than  next  the  water?  Note  that  the  heat  of  vaporiza- 
tion and  the  conductivity  are  given  in  C.G.S.  units. 


CHAPTER  XVII 
METEOROLOGY 

220.  General  Discussion. — Meteorology  is  that  science  which 
treats,  in  the  main,  of  the  variations  in  heat  and  moisture  of  the 
atmosphere,  and  the  production  of  storms  by  these  variations. 
Although  the  earth's  atmosphere  extends  to  a  height  of  a  great 
many  miles,  the  weather  is  determined  almost  entirely  by  the 
condition  of  the  lower,  denser  strata,  extending  to  a  height  of 
but  a  few  miles. 

Clouds. — Clouds  have  been  divided  into  eight  or  ten  important 
classes,  according  to  their  appearance  or  altitude.  Their  altitude 
varies  from  -1/2  mile  to  8  or  10  miles,  and  their  appearance  varies 
from  the  dense,  gray,  structureless  rain  cloud,  called  Nimbus,  to 
the  interesting  and  beautiful  "wool-pack"  cloud,  known  as 
Cumulus,  which  resembles  the  smoke  and  "steam"  rolling  up 
from  a  locomotive.  All  clouds  are  composed  either  of  minute 
droplets  of  water  or  tiny  crystals  of  snow,  floating  in  the  air. 
Fog  is  merely  a  cloud  at  the  surface  of  the  earth.  Thus,  what  is 
a  cloud  to  the  people  in  the  valley,  is  a  fog  to  the  party  on  the 
mountain  side  enveloped  by  the  cloud.  The  droplets  in  a  fog 
are  easily  seen.  The  upper  clouds  may  travel  in  a  direction 
quite  different  from  that  of  the  surface  wind,  and  at  velocities 
as  high  as  200  miles  per  hour. 

221.  Moisture  in  the  Atmosphere. — The  constant  evaporation 
from  the  ocean,  from  inland  bodies  of  water,  and  from  the  ground, 
provides  the  air  with  moisture,  the  amount  of  which  varies 
greatly  from  time  to  time.     Although  the  water  vapor  seldom 
forms  as  much  as  2  per  cent,  of  the  weight  of  the  air,  never- 
theless, water  vapor  is  the  most  important  factor  in  determining 
the  character  of  the  weather.     When  air  contains  all  of  the  mois- 
ture it  will  take  up,  it  is  said  to  be  saturated.     If  saturated  air 
is  heated,  it  is  capable  of  taking  up  more  moistufe;  while  if  it  is 
cooled,  it  precipitates  a  portion  of  its  moisture    as  fog,  cloud, 
dew,  or  rain.     If  still  further  cooled,  it  loses  still  more  of  its 
water  vapor.     Indeed  the  statement  that  the  air  is  saturated  with 

302 


METEOROLOGY  303 

water  vapor  does  not  indicate  how  much  water  vapor  it  contains, 
unless  the  temperature  of  the  air  is  also  given. 

When  unsaturated  air  is  cooled  more  and  more,  it  finally 
reaches  a  temperature  at  which  precipitation  of  its  moisture 
occurs.  This  temperature  is  called  the  Dew  Point.  If  air  is 
nearly  saturated,  very  little  cooling  brings  it  to  the  dew  point. 
After  the  dew  point  is  reached,  the  air  cools  more  slowly,  because 
every  gram  of  water  vapor  precipitated,  gives  up  nearly  600 
calories  of  heat  (its  heat  of  vaporization)  to  the  air.  Thus,  if 
on  a  clear  chilly  evening  in  the  fall,  a  test  for  the  amount  of 
water  vapor  in  the  air  shows  the  dew  point  to  be  several  degrees 
below  zero,  then  frost  may  be  expected  before  morning;  while 
if  the  dew  point  is  well  above  zero,  there  is  little  probability  of 
frost.  This  might  be  taken  as  a  partial  guide  as  to  whether  or 
not  to  protect  delicate  plants.  The  fact  should  be  emphasized, 
that  if  the  moisture  in  the  air  is  visible,  it  is  in  the  form  of  drop- 
lets, since  water  vapor,  like  steam,  is  invisible. 

222.  Hygrometry  and  Hygrometers. — Hygrometry  deals  with 
the  determination  of  the  amount  of  moisture  in  the  atmosphere. 
The  devices  used  in  this  determination  are  called  hygrometers. 
Only  two  of  these,  the  chemical  hygrometer  and  the  wet-and- 
dry-bulb  hygrometer,  will  be  discussed. 

The  Chemical  Hygrometer  consists  of  a  glass  tube  containing 
fused  calcium  chloride  (CaCl2),  or  some  other  chemical  having 
great  affinity  for  water.  Through  this  tube  (previously  weighed) 
a  known  volume  of  air  is  passed.  This  air,  during  its  passage, 
gives  up  its  moisture  to  the  chemical  and  escapes  as  perfectly 
dry  air.  The  tube  is  again  weighed,  and  the  gain  in  weight 
gives  the  amount  of  water  vapor  in  this  known  volume  of  air. 

The  Wet-and-dry-bulb  Hygrometer. — From  the  two  temperature 
readings  of  the  wet-and-dry-bulb  thermometer  (Sec.  198),  in 
connection  with  a  table  such  as  given  below,  the  dew  point 
may  be  found.  Having  found  the  dew  point,  the  amount  of 
moisture  per  cubic  yard  is  readily  found  from  the  second  table. 

The  manner  of  using  these  tables  will  be  best  illustrated  by  an 
example.  Suppose  that  when  a  test  is  made,  the  dry-bulb 
thermometer  reads  60°  F.,  and  the  wet-bulb  thermometer  52°  F., 
or  8°  lower.  Running  down  the  vertical  column  (first  table)  for 
which  i  —  t'  is  8°  until  opposite  the  dry-bulb  reading  60°,  we  find 
the  dew  point  45°.6  F.  This  shows  that  if  the  temperature  of  the 
air  falls  to  45°.6  F.,  precipitation  will  commence.  Opposite  to 


304 


MECHANICS  AND  HEAT 


dew  point  45°  (the  nearest  point  to  45.6)  in  the  second  table, 
we  find  0.299  and  0.0133;  which  shows  that  every  cubic  yard  of 
air  contained  approximately  0.0133  Ibs.  of  water  vapor  on  the 
day  of  this  test,  and  that  the  water  vapor  pressure  was  0.299 
inches  of  mercury. 

DEW  POINTS  FROM  WET-AND-DRY-BULB  HYGROMETER  READINGS 
Dry  bulb  temperature  t.     Wet  bulb  temperature  t'.     Difference  t  -t'. 


t°-r 


=  2° 


8°       |    10° 


12° 


14°  F. 


t° 

40°  F. 

36.2 

30.8 

25.6 

20.8 

16.0 

11.2 

6.4 

45 

41.4 

35.8 

31.2 

27.0 

22.1 

17.4 

12.8 

50 

45.8 

41.6 

37.4 

33.0 

29.0 

24.8 

20.6 

55 

51.0 

47.0 

43.0 

39.0 

35.2 

31.0 

27.0 

60 

56.4 

52.8 

49.2 

45.6 

42.3 

38.4 

34.8 

65 

61.6 

59.2 

55.0 

51.4 

48.0 

44.6 

41.0 

70 

67.0 

64.0 

61.0 

58.0 

55.3 

52.0 

49.0 

75 

72.0 

69.0 

66.0 

63.0 

60.6 

57.0 

54.0 

80 

77.0 

74.0 

71.0 

68.0 

65.0 

62.0 

59.0 

DEW  POINTS  AND  THE  CORRESPONDING  PRESSURES  AND 
DENSITIES  OF  WATER  VAPOR 


Dew  point 

Pressure  in  inches  of 
mercury 

Density  in  Ibs.  per  cu.  yd 

20°  F. 

0.102 

0.0051 

25 

0.130 

0.0066 

30 

0.161 

0.0076 

35 

0.200 

0.0091 

40 

0.252 

0.0111 

45 

0.299 

0.0133 

50 

0.358 

0.0159 

55 

0.433 

0.0196 

60 

0.512 

0.0223 

65 

0.617 

0.0255 

70 

0.732 

0.0306 

223.  Winds,  Trade  Winds. — Winds  originate  in  the  uneven 
heating  of  the  earth's  atmosphere  at  different  points.  This 
heating  is  in  part  due  to  the  direct  action  of  the  sun,  and  in  part 
to  the  heat  of  vaporization  given  off  when  a  portion  of  the  mois- 
ture in  the  air  changes  to  the  liquid  state.  When  air  is  heated 
it  expands,  and  therefore  becomes  lighter  and  rises  with  con- 
siderable velocity.  The  current  of  colder  air,  rushing  in  to  take 
its  place,  is  called  Wind.  This  effect  is  easily  noticed  with  a 
large  bonfire  on  a  still  day.  The  violent  upward  rush  of  the 
heated  air  above  the  fire  carries  cinders  to  a  great  height.  The 


j  METEOROLOGY  305 

cool  air  rushinp;  in  to  take  its  place  produces  a  "wind"  that 
blows  toward  the  fire  from  all  directions. 

The  Trade  Winds. — An  effect  similar  to  that  produced  by  the 
bonfire,  as  above  described,  is  constantly  being  produced  on  a 
grand  scale  in  the  tropical  regions.  The  constant  high  tempera- 
ture of  the  equatorial  regions  heats  the  air  highly  and  causes  it 
to  rise.  The  air  north  and  south  of  this  region,  rushing  in  to 
take  the  place  of  this  rising  air,  constitutes  the  Trade  Winds. 

On  account  of  the  rotation  of  the  earth,  trade  winds  do  not 
blow  directly  toward  the  equator  but  shift  to  the  westward. 
Thus  in  the  West  Indies,  the  trade  winds  are  N.  E.  winds,  i.e., 
they  blow  S.  W.  The  trade  winds  south  of  the  equator  are  S.  E. 
winds,  i.e.,  they  blow  N.  W. 

The  westward  deviation  of  the  trade  winds,  both  north  and 
south  of  the  equator,  may  be  accounted  for  as  follows.  Objects 
near  the  equator  describe  each  day,  due  to  the  rotation  of  the 
earth,  paths  which  are  the  full  circumference  of  the  earth; 
while  objects  some  distance  either  north  or  south  of  the  equator 
describe  shorter  paths  in  the  same  time,  and  therefore  have 
less  velocity.  Consequently,  as  a  body  of  air  moves  toward  the 
equator,  it  comes  to  points  of  higher  eastward  velocity,  and 
therefore  "falls  behind,"  so  to  speak;  that  is,  it  drifts  somewhat 
to  the  westward. 

Between  the  trade  winds  of  the  two  hemispheres  lies  the 
equatorial  "zone  of  calms."  This  zone,  which  varies  from 
200  to  500  miles  in  width,  has  caused  sailing  vessels  much  trouble 
with  its  prolonged  calms,  violent  thunder  storms,  and  sudden 
squalls. 

Since  a  rising  column  of  air  is  cooled  by  expansion,  it  precipi- 
tates its  moisture;  whereas  a  descending  column,  warmed  as  it 
is  by  compression,  is  always  capable  of  absorbing  more  moisture 
and  is,  therefore,  relatively  dry.  In  the  zone  of  calms,  the  air 
from  the  two  trade  winds  which  meet  in  this  region  must  rise. 
As  it  rises,  it  is  cooled  and  precipitates  its  moisture  in  torrents 
of  rain.  Wherever  the  prevailing  wind  blows  from  the  sea 
across  a  mountain  range  near  the  coast,  the  rain  will  be  ex- 
cessive on  the  mountain  slope  toward  the  sea,  where  the  air 
must  rise  to  pass  over  the  mountain.  As  the  air  descends  upon 
the  opposite  side  of  the  range,  it  is  very  dry  and  produces  a  region 
of  scant  rain  and,  in  many  cases,  a  desert.  The  rainfall  on  por- 
tions of  the  southern  slope  of  the  Himalaya  Mountains  is  about 


306  MECHANICS  AND 

30  ft.  per  year;  while  to  the  north  of  the  range  lie  large  arid  or 
semi-arid  districts. 

224.  Land  and  Sea  Breezes. — Near  the  seashore,  especially 
in  warm  countries,  the  breeze  usually  blows  toward  the  shore 
from  about  noon  until  shortly  before  sunset,  and  toward  the 
sea  from   about  midnight  until   shortly  before    sunrise.     The 
former  is  called  a  Sea  Breeze;  the  latter,  a  Land  Breeze. 

These  breezes  are  due  to  the  fact  that  the  temperature  of  the 
land  changes  quickly,  while  the  temperature  of  the  ocean  is 
nearly  constant  (Sec.  185).  Consequently,  by  noon,  the  air 
above  the  land  has  become  considerably  heated,  and  is  therefore 
less  dense  than  the  air  over  the  ocean.  This  heated  air,  there- 
fore, rises,  and  the  air  from  the  ocean,  rushing  in  to  take  its 
place,  is  called  the  sea  breeze.  The  rising  column  of  air  becomes 
cooler  as  it  rises,  and  flows  out  to  sea.  Thus,  air  flows  from 
sea  to  land  near  the  earth,  and  from  land  to  sea  in  the  higher 
regions  of  the  atmosphere.  Toward  midnight  the  land  and  the 
air  above  it  have  become  chilled.  This  chilled,  and  therefore 
dense  air  flows  out  to  sea,  as  a  land  breeze;  while  the  air  from 
the  ocean  flows  toward  the  land  in  the  higher  region.  It  will 
be  observed,  then,  that  the  convection  circulation  at  night  is 
just  the  reverse  of  the  day  circulation. 

It  is  observed  that  the  sea  breeze  first  originates  some  distance 
out  at  sea  and  blows  toward  the  land.  A  feasible  explanation 
is  this:  As  the  air  over  the  land  first  becomes  heated  it  expands 
and  swells  up  like  a  large  blister.  The  air  above,  lifted  by  the 
"blister,"  flows  away  out  to  sea  in  the  higher  regions  of  the 
atmosphere,  thereby  causing  an  excess  pressure  upon  the  air 
there.  The  air  then  flows  away  from  this  region  of  excess  pres- 
sure toward  the  land,  where  the  deficit  in  pressure  exists. 

225.  Cyclones. — Strictly,    the   term    cyclone   applies   to   the 
periodical  rotary  storms,  about  1000  miles  or  so  in  diameter, 
which  occur  in  various  parts  of  the  earth.     Every  few  days 
they  pass   across   the   central    portion    of    the    United  States, 
in  a   direction   somewhat    north   of  east.     Their   courses  may 
be  followed  from  day  to  day  by  means  of  the  U.  S.  weather 
maps.     The   barometric    pressure    is    usually    about    one-half 
inch  of  mercury  less  at  the  center  ("storm  center")  of  a  cyclone, 
than  at  the  margin.     This  region  of  low  pressure,  called  a  "low 
area,"    is  due,  at  least  in  part,  to  the  condensation  of  water 
vapor  that  occurs  in  cloud    formation,    and  the    consequent 


METEOROLOGY  307 

heating  of  the  air  by  the  heat  of  vaporization  thereby  evolved. 
This  heated  air  rises,  and  the  surrounding  air,  rushing  in, 
produces  wind.  Due  to  the  rotation  of  the  earth,  these  winds, 
instead  of  blowing  straight  toward  the  storm  center,  are,  in 
general,  deflected  to  the  right  of  the  storm  center.  Occasionally, 
due  so  some  local  disturbance,  the  wind  may  blow  in  a  direction 
nearly  opposite  to  that  which  would  be  expected  from  the  above 
rule;  but,  in  general,  the  surrounding  air  moves  toward  the  storm 
center  in  a  spiral  path.  The  rotation  is  counterclockwise 
(viewed  from  above)  in  the  Northern  hemisphere,  and  clockwise 
in  the  Southern.  Any  body  in  motion  (e.g.,  a  rifle  ball)  in  the 
Northern  hemisphere  tends  to  deviate  to  the  right  from  its  path, 
and  in  the  Southern  hemisphere  to  the  left.1  This  fact  accounts 
for  the  rotatory  motion  of  these  storms,  as  explained  below. 

Cause  of  the  Rotary  Motion  of  Cyclones. — Let  Fig.  145  represent 
a  top  view  of  a  level  table  A  upon  which  rests  a  heavy  ball  B 
loosely  surrounded  by  a  very  light  frame  C  to  which  is  attached 
a  string  DD\.  Evidently,  if  the  table  is  at  rest  and  the  lower  end 
of  the  string  Z>i,  which  passes  through  a  hole  in  the  center  of  the 
table,  is  pulled,  ball  B  will  roll  in  a  straight  path  to  the  center. 
If,  however,  the  string  is  pulled  while  the  table,  and  consequently 
the  ball,  are  rapidly  revolving  in  the  direction  indicated  by  the 
arrow  a,  then  the  ball  will  follow  a  left-handed  spiral  path  as  in- 
dicated by  the  broken  line.  For,  as  the  ball  moves  nearer  to 
the  center,  it  reaches  portions  of  the  table  of  smaller  and  smaller 
radius,  and  consequently  portions  having  less  tangential  velocity 
than  its  own.  Therefore,  the  ball  rolls  "ahead,"  i.e.,  to  the  right, 
of  the  straight  line  D  as  shown.  If  the  table  and  the  ball  were 
rotated  in  the  opposite  direction  (clockwise),  similar  reasoning 
would  show  that  the  ball  would  then  travel  toward  the  center  in 
a  right-handed  spiral  path. 

It  will  next  be  shown  that  any  area  of  the  globe,  having  a  diameter 
of,  say,  a  few  hundred  miles,  may  be  considered  to  be  a  flat  surface, 

Although  this  tendency  of  a  moving  body  to  drift  to  the  right  in  the 
Northern  hemisphere  and  to  the  left  in  the  Southern  hemisphere  is  of 
such  great  importance  in  determining  the  motion  of  storms,  its  effe.ct  on 
projectiles  is  very  slight  indeed.  Thus,  in  latitude  40°,  due  to  this  cause, 
an  army  rifle  projectile  veers  to  the  right  (no  matter  in  what  direction  it 
is  fired)  by  only  about  3  in.  on  a  1000-yd.  range.  Due  to  the  same  cause,  a 
heavy  locomotive,  when  at  full  speed  on  a  level  track,  bears  only  about 
50  Ibs.  more  on  the  right  rail  than  on  the  left. 


308 


MECHANICS  AND  HEAT 


rotating  about  a  vertical  axis  at  its  center,  and  that  consequently,  air 
which  tends  to  move  toward  the  center  of  the  area,  as  it  does  in  cyclones, 
will  trace  a  spiral  path  similar  to  that  traced  by  the  ball  (Fig.  145). 
That  an  area  about  the  north  pole  has  such  rotation,  with  the  pole  as 
axis,  is  evident.  Since  the  earth  rotates  from  west  to  east,  this  rotation 
viewed  from  above,  is  counterclockwise  the  same  as  shown  in  Fig. 
145.  The  rotational  velocity,  say  wi,  is  of  course  one  revolution  per 
day. 

Such  an  area  at  the  equator  would  revolve  once  a  day  about  a  hori- 
zontal (N.  and  S.)  axis,  but  would  obviously  have  no  rotation  about  a 


FIG.  145. 

vertical  axis.  This  fact  accounts  for  the  absence  of  cyclones  near  the 
equator. 

It  can  be  shown  that  such  an  area,  in  latitude  0,  has  an  angular 
velocity  o>  about  the  vertical  axis,  given  by  the  equation 

w  =  wi  sin  6 

The  rotation  of  the  area  is  counterclockwise  in  the  northern  hemisphere 
(see  rotation  at  the  north  pole  above)  and  clockwise  in  the  southern. 
Consequently,  the  air  moves  (i.e.,  the  wind  blows)  toward  the  center  of 
a  cyclone  in  a  left-handed  spiral  path  in  the  former  case,  and  in  a  right- 
handed  spiral  path  in  the  latter  case,  as  explained  above  for  the  ball. 

Hurricanes  and  Typhoons. — The  hurricanes  of  the  West  Indies,  and 
the  Typhoons  of  China,  might  be  called  the  "cyclones"  of  the  tropical 
and  sub-tropical  regions.  They  are  more  violent  and  of  smaller  diameter 
than  cyclones,  their  diameter  rarely  exceeding  400  miles,  though  they 
sometimes  gradually  change  to  cyclones  and  travel  long  distances  through 
the  temperate  zones. 


METEOROLOGY  309 

226.  Tornadoes. — Tornadoes  resemble  hurricanes,  but  are 
much  smaller,  and  usually  more  violent.  Because  of  the  terrific 
violence,  narrow  path,  brief  duration,  and  still  more  brief  warning 
given,  tornadoes  have  not  been  very  satisfactorily  studied,  and 
much  difference  of  opinion  exists  with  regard  to  them.  The 
visible  part  of  a  tornado  consists  of  a  depending,  funnel-shaped 
cloud,  tapering  to  a  column  which  frequently  extends  to  the 
ground.  Due  to  the  centrifugal  force  caused  by  the  rapid 
rotation  of  the  column,  the  air  pressure  within  it  is  considerably 
reduced.  Consequently  as  moist  air  enters  the  column  it  is 
cooled  by  expansion  and  its  moisture  condenses,  forming  the 
cloud  which  makes  the  column  visible.  At  sea,  tornadoes  are 
called  Water  Spouts.  The  column  is  not  water,  however,  but 
cloud  and  spray. 

Origin. — Tornadoes  usually  develop  to  the  southeast  of  the 
center  of  a  cyclone.  Sometimes  several  may  rage  simultaneously 
at  different  points  in  the  same  cyclone.  Occasionally  con- 
ditions of  the  atmosphere  arise  which  are  especially  favorable 
to  the  formation  of  tornadoes.  These  conditions  are  a  warm 
layer  of  air  saturated  with  moisture  next  to  the  earth,  with  a 
layer  of  much  cooler  air  above  it.  As,  due  to  local  disturbance, 
some  of  this  heated  moist  air  rises  to  the  cooler  regions,  it  pre- 
cipitates part  of  its  moisture,  thus  freeing  a  considerable  amount 
of  heat.  This  heat  prevents  the  rising  air  from  cooling  so  rapidly 
as  it  otherwise  would,  and  consequently  helps  to  maintain  its 
tendency  to  rise.  As  this  air  rises,  it  is  followed  by  other  satu- 
sated  air,  which  in  turn  receives  heat  by  condensation  of  its  water 
vapor.  Thus  the  action,  when  once  started,  continues  with 
great  violence.  The  air  rushing  in  from  the  surrounding  country 
to  take  the  place  of  the  ascending  air  current  acquires  a  rotary 
motion,  just  as  already  explained  in  connection  with  cyclones 
As  the  tornado  advances  it  is  constantly  furnished  with  a  new 
supply  of  hot,  damp  air  and  it  will  continue  just  so  long  as  this 
supply  is  furnished,  i.e.,  until  it  passes  over  the  section  of  country 
in  which  these  favorable  conditions  exist.  Tornadoes  travel 
across  the  United  States  in  a  direction  which  is  usually  about 
east.  A  tornado  may  be  likened  to  a  forest  fire,  in  that  the  one 
requires  a  continuous  supply  of  moist  air,  the  other,  a  continuous 
supply  of  fuel. 

Tornadoes  sometimes  do  not  reach  to  the  earth,  which  indicates 
that  the  favorable  stratum  of  air  upon  which  they  "feed"  is,  at 


310  MECHANICS  AND  HEAT 

least  sometimes,  at  a  considerable  altitude.  Some  think  it  is 
always  at  a  high  altitude.  This,  the  writer  doubts.  The  moist 
stratum  is  probably  very  deep. 

Extent. — The  destructive  paths  of  tornadoes  vary  in  width 
from  100  ft.  to  1/2  mile,  and  in  length  from  less  than  a  mile  to 
200  miles.  A  tornado  which  wrecks  weak  buildings  over  a 
path  1/8  mile  in  width  may  leave  the  ground  practically  bare 
for  a  width  of  100  ft.  or  so. 

Velocity. — The  velocity  of  tornadoes  varies  from  10  to  100 
miles  per  hour.  It  is  estimated,  however,  that  the  wind  near 
the  center  sometimes  attains  a  velocity  of  200  or  300  miles  per 
hour,  or  even  greater. 

Judging  by  the  effects  produced,  the  velocity  must  be  very 
great.  An  iron  bed  rail  has  been  driven  through  a  tree  by  a 
tornado.  A  thin-bladed  shovel  has  been  driven  several  inches 
into  a  tree.  Such  a  shovel  would  not  withstand  driving  into  a 
tree  with  a  sledge  hammer.  Splintered  boards  are  frequently 
driven  deep  into  the  ground,  and,  by  way  of  contrast,  mention 
may  be  made  of  a  ladder  which  was  laid  down,  at  a  considerable 
distance  from  the  path,  so  gently  as  to  scarcely  leave  a  mark  on 
the  ground.  Shingles  and  thin  boards  have  been  found  in  great 
numbers  6  or  7  miles  from  the  path,  and  probably  10  or  15  miles 
from  where  they  began  their  flight. 

The  rapid  rotational  velocity  at  the  center,  tends  to  produce 
a  vacuum,  as  already  mentioned.  It  is  conjectured  that  the 
pressure  at  the  center  of  the  tornado  may  be  as  much  as  3  or  4 
Ibs.  per  square  inch  less  than  normal.  If  this  be  true,  then,  as 
the  tornado  reaches  a  building  filled  with  air  at  nearly  normal 
pressure,  there  will  be  an  excess  pressure  within  the  building  of 
say  3  Ib.  per  square  inch,  or  over  400  Ibs.  per  square  foot,  tending 
to  make  the  building  explode.  The  position  of  the  wreckage 
sometimes  indicates  that  this  is  just  what  has  taken  place. 

In  spite  of  the  great  violence  of  tornadoes,  few  people  are  killed 
by  them,  because  of  their  infrequency  and  limited  extent.  If  a 
man  were  to  live  a  few  hundred  thousand  years  he  might  reason- 
ably expect  to  be  caught  in  the  path  of  a  tornado,  and  if  immune 
from  death  except  by  tornadoes  he  could  not  reasonably  expect 
to  live  more  than  a  few  million  years. 


CHAPTER  XVIII 
STEAM  ENGINES  AND  GAS  ENGINES 

227.  Work  Obtained  from  Heat — Thermodynamics. — Thermo- 
dynamics deals  with  the  subject  of  the  transformation  of  heat 
into  mechanical  energy,  and  vice  versa,  and  the  relations  that 
obtain  in  such  transformation  under  different  conditions.  No 
attempt  will  be  made  to  give  more  than  a  brief  general  treatment 
of  this  important  subject. 

What  is  known  as  the  First  Law  of  Thermodynamics  may  be 
stated  as  follows:  Heat  may  be  transformed  into  mechanical 
energy,  and  likewise,  mechanical  energy  may  be  transformed  into 
heat,  and  in  all -cases,  the  ratio  of  the  work  done,  to  the  heat  so 
transformed,  is  constant.  Conversely,  the  ratio  of  the  work 
supplied,  to  the  heat  developed  (in  case  mechanical  energy  is 
changed  to  heat  energy  by  friction,  etc.),  gives  the  same  constant. 
This  constant,  in  the  British  system,  is  778.  Thus,  if  one  B.T.U. 
of  heat  is  converted  into  mechanical  energy,  it  will  do  778  ft.-lbs. 
of  work;  conversely,  if  778  ft.-lbs.  of  work  is  converted  into  heat, 
it  produces  one  B.T.U.  For  example,  if  778  ft.-lbs.  of  energy  is 
used  in  stirring  1  Ib.  of  water,  it  will  warm  the  water  1°  F.  The 
similar  relation  in  the  metric  system  is  expressed  by  the  state- 
ment that  1  calorie  equals  4.187X107  ergs. 

Illustrations  of  the  First  Law  of  Thermodynamics. — By  means  of 
the  steam  engine  and  the  gas  engine,  heat  is  converted  into  mechan- 
ical energy.  In  bringing  a  train  to  rest,  its  kinetic  (mechanical) 
energy  is  converted  into  heat  by  the  brakes,  where  a  shower  of 
sparks  may  be  seen.  In  inflating  a  bicycle  tire,  work  is  done  in 
compressing  the  air,  and  this  heated  air  makes  the  tube  leading 
from  the  pump  to  the  tire  quite  warm.  In  the  fire  syringe,  a 
snug-fitting  piston,  below  which  some  tinder  is  fastened,  is 
quickly  forced  into  a  cylinder  containing  air.  As  the  air  is  com- 
pressed it  is  heated  sufficiently  to  ignite  the  tinder.  In  gas  en- 
gines, preignition  may  occur  during  the  compression  stroke,  due 
in  part  to  the  heat  developed  by  the  work  of  compression. 

311 


312  MECHANICS  AND  HEAT 

The  Second  Law  of  Thermodynamics. — The  second  law  of 
thermodynamics  is  expressed  by  the  statement  that  heat  will 
not  flow  of  itself  (i.e.,  without  external  work),  from  a  colder  to  a 
warmer  body.  In  the  operation  of  the  ammonia  refrigerating 
apparatus,  heat  is  taken  continuously  from  the  very  cold  brine 
and  given  to  the  very  much  warmer  cooling  tank;  but  the  work 
required  to  cause  this  "uphill"  flow  of  heat  is  done  by  the  steam 
engine  which  operates  the  air  pump. 

Lord  Kelvin's  statement  of  the  second  law  amounts  to  this: 
Work  cannot  be  obtained  by  using  up  the  heat  in  the  coldest 
bodies  present.  Carnot  (Sec.  236)  showed  that  when  heat  passes 
from  a  hotter  to  a  colder  body  (through  an  engine)  the  maximum 

m    rp 

fraction  of  the  heat  which  may  be  converted  into  work  is  — ^ — 2' 

in  which  TI  and  772  are,  respectively,  the  temperatures  of  the 
two  bodies  on  the  absolute  scale. 

228.  Efficiency. — While  it  is  possible  to  convert  mechanical 
energy,  or  work,  entirely  into  heat,  thereby  obtaining  100  per 
cent,  efficiency,  it  is  impossible  in  the  reverse  process  to  trans- 
form more  than  a  small  percentage  of  heat  energy  into  mechanical 
energy.  It  is,  indeed,  a  very  good  steam  engine  that  changes 
into  work  1/5  of  the  heat  energy  of  the  steam  furnished  it  by 
the  boiler.  Considering  the  large  amount  of  heat  that  radiates 
from  the  furnace,  and  also  the  heat  that  escapes  through  the 
smoke  stack,  there  is  a  further  reduction  in  the  efficiency.  The 
total  efficiency  of  a  steam  engine  is  the  product  of  three  efficien- 
cies; that  of  the  furnace,  that  of  the  boiler,  and  that  of  the  engine. 

The  furnace  wastes  about  1/10  of  the  coal  due  to  incomplete 
combustion,  through  escape  of  unburnt  gases  up  the  smoke 
stack,  and  unburnt  coal  into  the  ash  pit.  The  furnace  efficiency 
is,  therefore,  about  9/10  or  90  per  cent.  About  4/10  of  the  heat 
developed  by  the  furnace  escapes  into  the  boiler  room  or  up  the 
smoke  stack;  so  that  the  boiler  efficiency  is  about  6/10  or  60  per 
cent.  A  good  "condensing"  engine  converts  into  work  about 
1/5  of  the  heat  energy  furnished  it  by  the  boiler,  in  other  words, 
its  efficiency  is  about  20  per  cent.  The  total  efficiency  E  of 
the  steam  engine,  which  may  be  defined  by  the  equation 

work  done 


energy  of  fuel  burned 
has,  then,  the  value  1/5X6/10X9/10,  or  about  11  per  cent. 


STEAM  ENGINES  AND  GAS  ENGINES          313 

Calculation  of  Efficiency. — The  efficiency  of  the  steam  engine 
varies  greatly  with  the  care  of  the  furnace,  and  the  type  and 
size  of  the  boiler  and  engine.  Few  engines  have  a  total  efficiency 
above  12  per  cent.,  and  many  of  the  smaller  ones  have  as  low  as 
4  or  5  per  cent,  efficiency,  or  even  lower.  Coal  which  has  a  heat 
of  combustion  of  14,000  B.T.U.  per  lb.,  contains  14,000X778  ft.- 
Ibs.  of  energy  per  pound.  One  H.P.-hr.  is  3600X550  ft.-lbs. 
,.  ,  3600X550  .,  .  .  .  t  ..  .,  ,  . 

Accordingly  ^QQQ  x  773  *".,   or  approximately   1/5  lb.  of  coal 

would  do  1  H.P.-hr.  of  work  if  the  efficiency  of  the  engine  were 
100  per  cent.  If  an  engine  requires  4  Ibs.  of  coal  per  H.P.- 
hr.,  its  efficiency  is  approximately  1/4X1/5,  or  5  per  cent.  In 
order  to  make  an  accurate  determination  of  the  efficiency,  the 
heat  of  combustion  would  have  to  be  known  for  the  particular 
grade  of  coal  used. 

Limiting,  or  Thermodynamic  Efficiency. — Carnot  (Sec.  236) 
showed  that  the  efficiency  of  an  ideal  engine,  which,  of  course, 
cannot  be  surpassed,  is  determined  by  the  two  extreme  tempera- 
tures of  the  working  fluid  (steam  or  gas) .  If  heat  (say  in  steam) 
is  supplied  to  the  engine  at  127°  C.  or  400°  A.,  and  the  engine 
delivers  it  to  the  condenser  at  27°  C.,  or  300°  A.,  then  the  maxi- 

..     ,     ~  .  .    400-300 

mum  theoretical  efficiency  is  — TQ~ — ,  or  25  per  cent.  Ob- 
viously, then,  a  gain  in  efficiency  is  obtained  by  using  steam  at 
a  very  high  pressure,  and  consequently  at  a  high  temperature. 
The  high  efficiency  of  the  gas  engine  is  due  partly  to  the  great 
temperature  difference  employed,  and  partly  to  the  fact  that 
the  "furnace"  is  in  the  cylinder  itself,  thereby  reducing  heat 
"losses." 

Some  gas  engines  (Sec.  237)  have  more  than  30  per  cent,  ef- 
ficiency. Gas  engines  are  usually  more  troublesome  than  steam 
engines  and  also  less  reliable  in  their  operation;  nevertheless, 
because  of  their  greater  economy  of  fuel,  they  are  coming  into 
very  general  use. 

The  lightness  of  gas  engines  recommends  them  for  use  on 
automobiles,  motorcycles,  and  flying  machines.  Engines  weigh- 
ing about  2.5  Ibs.  per  H.P.  have  been  made  for  use  on  aero- 
planes. Indeed,  the  lightness  of  the  gas  engine  has  made 
possible  the  development  of  the  aeroplane. 

229.  The    Steam   Engine. — A   modern    steam    engine,    fully 


314 


MECHANICS  AND  HEAT 


equipped  with  all  of  its  essential  attachments,  is  a  very  compli- 
cated mechanism. 

In  order  to  bring  out  more  clearly  the  fundamental  principles 
involved  in  the  action  of  the  steam  engine,  it  seems  best  to  omit 
important  details  found  in  the  modern  engine,  since  these  de- 
tails are  confusing  to  the  beginner,  and  therefore  serve  to  obscure 
the  underlying  principles.  In  accordance  with  this  idea,  an  ex- 
ceedingly primitive  engine  is  shown  in  Fig.  146.  In  Fig.  147, 
an  engine  is  shown  which  is  essentially  modern,  although  certain 
details  of  construction  are  purposely  omitted  or  modified,  espe- 
cially in  the  indicator  mechanism,  and  in  the  valve  mechanism. 

In  Fig.  146,  A  is  a  pipe  which  carries  steam  from  the  boiler  to 
the  cylinder  B,  through  either  valve  a  or  valve  b,  depending  upon 
which  is  open.  P  is  the  piston,  and  C  is  the  piston  rod,  which 


FIG.  146. 

passes  through  the  end  of  the  cylinder  (through  steam-tight 
packing  in  a  "stuffing  box")  to  the  crosshead  D.  As  the  piston 
is  forced  back  and  forth  by  the  steam,  as  will  be  explained  below, 
the  crosshead  moves  to  and  fro  in  "guides,"  indicated  by  the 
broken  lines.  The  crosshead,  by  means  of  the  connecting  rod 
E  attached  to  the  crank  pin  F,  causes  the  crank  G  to  revolve  as 
indicated.  The  crank  G  revolves  the  crank  shaft  0,  to  which  is 
usually  attached  a  very  heavy  fly  wheel  H  in  order  to  "steady" 
the  motion. 

If  valves  a  and  c  are  open,  and  b  and  d  closed,  the  steam  passes 
from  the  boiler  into  the  cylinder,  and  forces  P  to  the  right. 
The  exhaust  steam  to  the  right  of  P  (remaining  from  a  former 
stroke)  is  driven  out  through  c  to  the  air.  When  P  reaches  the 


STEAM  ENGINES  AND  GAS  ENGINES          315 

right  end  of  the  cylinder,  valves  a  and  c  are  closed,  and  6  and  d 
are  opened,  thus  permitting  steam  to  enter  at  6  and  force  P  to  the 
left  end  again;  whereupon  the  entire  operation  is  repeated. 
These  valves  are  automatically  opened  and  closed  at  just  the 
right  instant  by  a  mechanism  connected  with  the  crank  shaft 
(Sec.  233).  In  practice,  valve  a  would  be  closed  when  P  had 
traveled  to  the  right  about  1/3  the  length  of  the  cylinder  (Sec. 
231). 

Speed  Regulation. — A  Centrifugal  Governor,  driven  by  the 
engine,  controls  the  steam  supply,  and  hence  the  speed,  by  open- 
ing wider  the  throttle  valve  (valve  not  shown)  in  A  if  the  speed 
is  too  low,  and  by  partially  closing  it  when  the  speed  is  too  high, 
as  explained  in  Sec.  63.  It  may  be  mentioned  that  some  gover- 
nors control  the  speed  by  regulating  the  cut-off  (Sec.  231);  that 
is,  by  admitting  steam  to  the  cylinder  during  a  small  fraction  of 
the  stroke,  in  case  the  speed  becomes  too  high. 

Compound  and  Triple  Expansion  Engines. — In  the  Compound 
Engine,  the  exhaust  steam  from  cylinder  B  passes  through  pipe 
/  to  a  second  cylinder,  where  it  drives  the  piston  to  and  fro, 
just  as  the  steam  from  pipe  A  drives  the  piston  shown  in  the 
figure.  If  the  exhaust  steam  from  this  second  cylinder  operates 
a  third  cylinder  we  have  a  Triple  Expansion  Engine — so-called 
because  the  steam  expands  three  times.  Obviously,  because  of 
this  expansion,  the  second  cylinder  must  be  larger  than  the  first, 
and  the  third  larger  than  the  second.  By  using  steam  at  very 
high  pressure  (about  200  Ibs.  per  sq.  in.),  and  expanding  it 
successively  in  there  different  cylinders,  a  much  higher  efficiency 
is  obtained  than  with  a  single-cylinder  engine.  It  will  be  evi- 
dent, that  the  more  the  steam  condenses  on  the  walls  of  the 
cylinder,  the  more  rapidly  its  pressure  drops  with  expansion.  It 
may  be  mentioned  that  the  greater  efficiency  of  the  triple  expan- 
sion engine  is  due  principally  to  a  reduction  of  this  condensation. 

Superheating. — Another  method  of  reducing  condensation  is 
to  superheat  the  steam.  If  the  steam  is  conducted  from  the 
boiler  to  the  engine  through  coiled  pipes  surrounded  by  moder- 
ately hot  flame,  it  may  thereby  have  its  temperature  raised  as 
much  as  200°  F.,  and  is  then  said  to  be  superheated  200°.  Super- 
heated steam  does  not  so  readily  condense  upon  expansion  in  the 
engine  as  does  ordinary  steam,  and  consequently  gives  a  higher 
efficiency. 

Increasing  the  Efficiency. — The  efficiency  of  the  steam  engine 


316  MECHANICS  AND  HEAT 

has  been  increased,  step  by  step,  by  means  of  various  improve- 
ments, prominent  among  which  are,  the  expansive  use  of  steam 
in  the  cylinder  (Sec.  231),  the  expansion  from  cylinder  to  cylinder 
as  in  triple  expansion  engines,  and  the  condensation  of  the  ex- 
haust steam  ahead  of  the  piston  (Sec.  230)  to  eliminate  "back 
pressure."  To  these  may  be  added  the  use  of  higher  steam  pres- 
sure, and  also  the  use  of  superheated  steam. 

230.  Condensing  Engines. — It  will  be  observed,  that  in  the 
above   noncondensing  engine,  the  steam  from  the  boiler  has  to 
force  the  piston  against  atmospheric  pressure  (15  Ibs.  per  sq.  in). 
By  leading  the  exhaust  pipe  /  to  a  "condenser,"  which  condenses 
most  of  the  steam,  this  "back  pressure"  is  largely  eliminated. 
The  Condenser  consists  of  an  air-tight  metal  enclosure,  kept  cool 
either  by  a  water  jet  playing  inside,  or  by  cold  water  circulating 
on  the  outside.     The  former  is  called  the  Jet  Condenser  and  the 
latter,  the  Surface  Condenser.     A  pipe  from  an  air  pump  leads  to 
the  condenser,  and  by  means  of  this  pipe,  the  air  pump  removes 
the  water  and  air,  maintaining  in  the  condenser  a  fairly  good 
vacuum.     Assuming  that  the  boiler  pressure  is,  say,  60  Ibs.  per 
sq.  in.  (i.e.,  60  Ibs.  per  sq.  in.  above  atmospheric  pressure),  and 
that  the  condenser  maintains  in  the  cylinder,    "ahead"  of  the 
piston,  a  partial  vacuum  of  2  Ibs.  per  sq.  in.  pressure;  it  will  be 
evident  that  the  available  working  pressure  will  be  increased 
to  73  Ibs.  per  sq.  in.  (15-2=  13,  and  60+13  =  73),  and  that  there- 
fore the  efficiency  will  be  increased  in  about  the  same  ratio. 

231.  Expansive  Use  of  Steam,  Cut-off  Point. — If,  when  the 
piston  shown  in  Fig.  146  has  moved  to  the  right  1/4  the  length 
of  the  cylinder,  i.e.,  when  it  is  at  1/4  stroke,  valve  a  is  closed, 
then  only  1/4  as  much  steam  will  be  used  as  would  have  been 
used  had  valve  a  remained  open  to  the  end  of  the  stroke.     But  the 
work  done  by  the  piston  during  the  stroke  will  be  more  than  1/4 
as  much  in  the  first  case  as  in  the  second,  hence  steam  is  econo- 
mized.    If  a  is  closed  at  1/4  stroke,  the  Cut-off  Point  is  said  to 
be  at  1/4  stroke. 

The  work  done  per  stroke,  if  the  valve  a  remains  open  during 
the  full  stroke,  is  FL  (or  Fd,  since  Work=Fd),  in  which  F  is  the 
force  exerted  on  the  piston  (its  area  A  times  the  steam  pressure 
p),  and  L  is  the  length  of  the  stroke.  Consequently,  the  work 
done  per  stroke  is  pAL.  If  the  cut-off  is  set  at  1/4  stroke,  then 
the  full  pressure  is  applied  for  the  first  quarter  stroke  only,  and 
therefore  the  work  done  by  the  steam  in  this  quarter  stroke 


STEAM  ENGINES  AND  GAS  ENGINES          317 

is  pAL/4.  During  the  remaining  3/4  stroke,  the  enclosed  steam 
expands  to  four  times  as  great  volume,  and  because  of  the  cool- 
ing effect  of  expansion,  it  has  its  pressure  reduced  at  the  end  of 
the  stroke  to  less  than  p/4,  the  value  which  Boyle's  Law  would 
indicate.  Assuming  the  average  pressure  during  the  last  3/4 
stroke  to  be  even  as  low  as  p/3,  we  have  for  the  work  of  this  3/4 
stroke 


We  see,  then,  that  by  using  the  expansive  power  of  the  steam 
during  3/4  of  the  stroke,  we  obtain  the  work  \  pAL,  which, 
added  to  |  pAL  obtained  from  the  first  1/4  stroke,  gives 
\  pAL  for  the  total  work.  But  the  work  obtained  per  stroke  by 
keeping  the  valve  a  open  during  the  full  stroke  was  pAL.  Hence 
the  total  work  per  stroke,  using  the  cut-off,  is  1/2  as  great  as 
without,  and  the  steam  consumption  is  only  1/4  as  great;  there- 
fore, the  Efficiency  is  doubled,  in  this  instance,  by  the  use  of  the 
cut-off. 

232.  Power.  —  Since  power  is  the  rate  of  doing  work,  or,  in 
the  units  usually  employed,  the  amount  of  work  done  per  second 
(Sec.  81),  it  will  be  at  once  evident  that  the  product  of  the  work 
per  stroke,  or  PAL  (Sec.  231),  and  the  number  of  strokes  (to  the 
right)  per  second,  or  N,  will  give  the  power  developed  by  the 
steam,  which  enters  at  the  left  end  of  the  cylinder.  That  is, 
power  —PALN,  in  which  P  is  the  average  difference  in  pressure 
upon  the  two  sides  of  the  piston  during  the  entire  stroke.  The 
average  pressure  is  easily  found  from  the  indicator  card  (Sec. 
234).  As  an  aid  to  the  memory,  the  symbols  may  be  rearranged 
so  as  to  spell  the  word  PLAN. 

If  P  is  expressed  in  pounds  per  square  inch,  and  A  in  square 
inches,  then  the  average  force  PA  exerted  by  the  piston  will  be  in 
pounds.  If  L  is  the  length  of  the  cylinder  in  feet,  then  PAL,  the 
work  done  per  stroke,  will  be  expressed  in  foot-pounds.  Finally, 
since  N,  the  number  of  revolutions  per  second,  is  also  the  number 
of  strokes  to  the  right  per  second,  the  power,  PLAN,  developed 
by  the  left  end  of  the  cylinder,  is  given  in  foot-pounds  per  second. 
Dividing  this  by  550  gives  the  result  in  horse  power;  i.e., 


H.P.  (one  end)  =  (87) 


318  MECHANICS  AND  HEAT 

If  N  represents  the  speed  of  the  engine  in  revolutions  per  minute 
(R.P.M.),  then,  since  1  H.  P.  -33000  ft.-lbs.  per  min.,  we  have 


H.P.  (one  end)  =  (87a) 


If  the  cut-off  point  for  the  stroke  to  the  left  does  not  occur 
at  exactly  the  same  fraction  of  the  stroke  as  it  does  for  the 
stroke  to  the  right,  then  the  average  pressure  pushing  the  pis- 
ton to  the  left  will  not  be  the  same  as  that  pushing  it  to  the  right, 
and  the  power  developed  by  the  right  end  of  the  cylinder  will 
differ  from  that  developed  by  the  left  end.  This  difference  usu- 
ally amounts  to  but  a  few  per  cent,  of  the  total  power. 

233.  The  Slide  Valve  Mechanism.—  The  slide  valve  V  (Fig. 
147)  is  operated  by  what  virtually  amounts  to  a  crank  of  length 
00',  called,  however,  an  Eccentric.  The  eccentric  consists  of  a 
circular  disc  «/.  whose  center  is  at  0',  attached  to  the  crank  shaft 
whose  center  is  at  0.  Over  J  passes  the  strap  K  connected  with 
the  eccentric  rod  L.  As  the  crank  shaft  revolves  clockwise,  0', 
which  is  virtually  the  right  end  of  rod  L,  moves  in  the  small 
dotted  circle  as  indicated.  This  circular  motion  causes  L  to 
move  to  and  fro,  thus  imparting  to  the  slide  valve  V  a  to-and-fro 
motion.  By  adjusting  the  eccentric  until  the  angle  between 
00'  and  the  crank  G  has  the  proper  value,  the  valve  opens  and 
closes  the  ports  at  the  proper  instants. 

With  the  valve  in  the  position  shown,  the  steam  from  the 
boiler,  entering  the  steam  chest  S  through  pipe  A,  passes  through 
steam  port  a  into  the  cylinder.  The  exhaust  steam,  from  the 
preceding  stroke,  escapes  through  steam  port  6  and  exhaust  port 
c  into  the  exhaust  pipe,  which  conducts  the  steam  in  the  direc- 
tion away  from  the  reader  to  the  condenser  (not  shown).  An 
instant  later,  port  a  will  be  closed  (cut-off  point),  and  the  steam 
then  in  the  left  end  of  the  cylinder  will  expand  (expansion  period, 
Sec.  231)  and  push,  the  piston  to  the  right.  As  the  piston  ap- 
proaches the  right  end,  the  valve  V  will  close  port  b  and  at  the 
same  time  open  port  a  into  the  exhaust  port  c.  This  releases  the 
steam  in  the  left  end  of  the  cylinder,  and  is  called  the  release  point. 
Since  b  is  closed  before  the  piston  reaches  the  right  end  of  its 
stroke,  there  still  remains  some  exhaust  steam  in  the  right  end  of 
the  cylinder.  This  steam  acts  as  a  "cushion"  and  reduces  the  jar- 
ring. During  the  last  part  of  the  stroke,  then,  the  piston  is 
compressing  exhaust  steam.  This  is  called  the  compression 


STEAM  ENGINES  AND  GAS  ENGINES 


319 


period.  About  the  time  the  piston  reaches  the  right  end  of  its 
stroke,  valve  V  has  moved  far  enough  to  the  left  to  open  port 
b  to  the  steam  chest,  thus  admitting  "live"  steam  to  the  right 
end  of  the  cylinder,  and  the  return  stroke,  similar  in  all  respects 
to  the  one  we  have  just  described,  occurs. 


234.  The  Indicator. — The  essentials  of  the  indicator  are  shown 
in  Fig.  147  (left  upper  corner).  /  is  a  small  vertical  cylinder 
containing  a  piston  N,  and  is  connected  by  pipes  with  the  ends 
of  the  engine  cylinder,  as  shown.  If  valve  e  is  closed  and  valve 
d  is  open,  it  will  be  evident  that,  as  the  pressure  in  the  left  end 
of  the  cylinder  rises  and  falls,  the  piston  N,  which  is  held  down 
by  the  spring  s,  will  rise  and  fall,  and  cause  the  pencil  p  at  the  end 
of  the  lever  Q  to  rise  and  fall. 

M  is  a  drum,  to  which  is  fastened  a  "card"  W.  This  drum  is 
free  to  rotate  about  a  vertical  axis  when  the  cord  T,  passing  over 
pulley  U,  is  pulled  to  the  right.  As  the  pull  on  T  is  released,  a 
spring  (not  shown)  causes  the  drum  to  rotate  in  the  reverse 
direction. 

It  will  therefore  be  seen  that  the  to-and-fro  (horizontal) 
motion  of  the  crosshead  D,  by  means  of  lever  R  and  string  T, 
causes  the  drum  to  rotate  to  and  fro,  and  consequently  move 
the  card  to  and  fro  under  the  pencil  p.  If  the  pencil  were  station- 
ary it  would  trace  a  straight  horizontal  line  on  the  card. 


320  MECHANICS  AND  HEAT 

Thus  we  see  that  the  change  of  pressure  in  the  cylinder  causes 
the  pencil  to  move  up  and  down,  while  the  motion  of  the  drum 
causes  the  card  to  move  horizontally  under  the  pencil.  In  prac- 
tice, both  of  these  motions  take  place  simultaneously,  and  the 
pencil  traces  over  and  over  the  curve  shown.  It  will  be  seen 
that  the  motion  of  the  card  under  the  pencil  exactly  reproduces, 
on  a  reduced  scale,  the  motion  of  the  piston  and  crosshead. 
That  is  to  say,  when  the  piston  P  has  moved  to  the  right,  say  1/4 
the  length  of  the  cylinder,  or  is  at  "quarter  stroke,"  the  pencil 
p  is  1/4  way  across  the  indicator  card,  and  so  on. 

The  indicator  card  is  shown,  drawn  to  a  larger  scale,  in  the 
upper,  right  corner  of  Fig.  147.  At  the  instants  that  the  piston, 
in  moving  to  the  right,  passes  points  1,  2,  3,  4,  the  pencil  p 
traces  respectively,  the  corresponding  points  1,  2,  3,  4,  on  the 
indicator  card.  As  the  piston  moves  back  to  the  left  from 
4  to  5,  pencil  p  traces  from  4  to  5  on  the  curve.  The  indicator 
card  shows  that  full  steam  pressure  acts  on  P  during  its  motion 
from  1  to  2;  that  at  2  the  inlet  valve  at  the  left  closes  (i.e.,  cut- 
off occurs,  see  slide  valve,  Sec.  233) ;  and  that  the  pressure  of 
the  enclosed  steam,  as  it  expands  and  pushes  the  piston  through 
the  remainder  of  the  stroke,  decreases,  as  indicated  by  the  points 
2,  3,  and  4. 

As  the  piston,  on  the  return  stroke,  reaches  the  point  marked 
5,  port  a  is  closed  and  the  compression  period  (Sec.  233)  begins. 
This  is  shown  on  the  indicator  diagram  by  the  rounded  corner 
at  5.  At  the  point  marked  6,  steam  is  again  admitted  through 
port  a,  and  the  pencil  p  rises  to  point  1  on  the  diagram.  The 
different  periods  shown  on  the  indicator  diagram  are,  then,  ad- 
mission of  steam  from  1  to  2,  expansion  from  2  to  4,  exhaust 
from  4  to  5,  and  compression  from  5  to  6. 

If  the  back  pressure  of  the  exhaust  steam  were  entirely  elimi- 
nated by  the  condenser,  the  pencil  on  the  return  stroke  would 
trace  a  lower  line  than  4-5,  say,  ii' .  The  distance  j  is  then  a 
measure  of  the  back  pressure,  which  would  be  about  2  or  3 
Ibs.  per  sq.  in.  when  using  a  condenser,  and  about  15  Ibs.  per  sq. 
in.  without  a  condenser. 

To  obtain  the  indicator  diagram  for  the  other  end  of  the 
cylinder  (shown  in  broken  lines  in  the  figure),  valve  d  is  closed 
and  valve  e  is  opened.  This  curve  should  be  (frequently  it  is 
not)  a  duplicate  of  the  curve  just  discussed,  in  the  same  sense 
that  the  right  br\nd  is  a  duplicate  of  the  left. 


STEAM  ENGINES  AND  GAS  ENGINES          321 

Use  of  the  Indicator  Card. — The  indicator  card  enables  the 
operator  to  tell  whether  the  engine  is  working  properly;  e.g., 
whether  the  admission  or  the  cut-off  are  premature  or  delayed, 
requiring  valve  adjustment;  or  whether  or  not  the  "back  pres- 
sure" is  excessive  due  to  fault  of  the  condenser,  and  so  on. 

Another  use  of  the  indicator  card  is  in  determining  the  average 
working  pressure  which  drives  the  piston.  By  subjecting  the 
indicator  piston  to  known  changes  of  pressure  as  read  by  a  steam 
gauge,  we  may  easily  determine  how  many  pounds  pressure  per 
square  inch  corresponds  to  an  inch  rise  of  the  pencil  p.  Having 
thus  calibrated  the  indicator,  suppose  we  find  that  an  increase  of 
40  Ibs.  per  sq.  in.  causes  p  to  rise  1  in.  Let  the  vertical  dotted 
line  through  3  across  the  indicator  curve  be  1 . 5  in.  in  length.  We 
then  know  that  at  1/2  stroke  the  available  working  pressure  on 
the  piston,  or  the  difference  between  the  pressure  on  the  left  and 
the  exhaust  pressure  on  the  right  side  of  the  piston,  is  60  Ibs. 
per  sq.  in.  Further,  suppose  that  when  we  divide  the  total  area 
of  the  curve  by  its  horizontal  length  we  obtain  2  in.  for  its 
average  height.  We  then  know  that  the  average  working  pressure 
P  for  the  entire  stroke  is  80  Ibs.  per  sq.  in.  This  average  value 
of  p,  thus  found,  is  the  P  of  Eq.  87,  which  gives  the  horse  power 
(H.P.)  of  the  engine. 

Since  the  average  height  of  the  indicator  diagram  gives  the 
average  working  pressure  on  the  piston,  and  since  its  length  is 
proportional  to  the  length  of  the  stroke  of  the  piston,  we  see  that 
its  area  is  proportional  to,  and  is  therefore  a  measure  of,  the 
work  done  per  stroke,  and  hence  a  measure  of  the  power.  Ac- 
cordingly, any  adjustment  of  valves  or  other  change  which  in- 
creases this  area  without  altering  the  speed,  produces  a  propor- 
tional increase  in  power.  If,  further,  the  same  amount  of  steam 
is  used  as  before,  then  there  is  a  proportional  increase  in  efficiency. 

235.  The  Steam  Turbine. — In  recent  years,  some  large  and 
very  efficient  steam  turbines  have  been  installed.  Because  of 
their  freedom  from  jarring,  which  is  so  great  in  the  reciprocating 
steam  engines,  and  also  because  of  their  high  speed,  they  are 
being  used  more  and  more  for  steamship  power. 

In  the  steam  turbine,  a  stream  of  steam  impinges  against 
slanting  vanes  and  makes  them  move  just  as  air  makes  windmill 
vanes  move  (Sec.  149).  It  differs  from  the  windmill,  however, 
in  that  the  stream  of  steam  must  be  confined,  just  as  water  is  in 
the  water  turbine.  Note  that  the  windmill  might  be  called  an 
21 


322 


MECHANICS  AND  HEAT 


air  turbine.  The  steam  turbine  differs  from  the  windmill  also 
in  that  each  portion  of  steam  must  pass  successively  several 
movable  vanes  alternating  with  fixed  vanes,  as  indicated  in 
Fig.  148.  The  rotor  vanes,  attached  to  the  rotating  part  called 
the  rotor,  are  indicated  by  heavy  curved  lines.  The  stator  vanes 
are  stationary  and  are  attached  to  the  tubular  shell  which  sur- 
rounds the  rotor  and  confines  the  steam.  The  stator  vanes  are 
indicated  in  the  sketch  by  the  light  curved  lines.  It  will  be 
understood  that  the  reader  is  looking  toward  the  axis  of  the 
rotor,  which  is  indicated  by  the  horizontal  line. 

As  the  steam   passes   to   the   right,  the   fixed   vanes   deflect 
it  somewhat  downward,  and  the  movable  vanes,  somewhat  up- 


FIG.  148. 


ward,  as  indicated  by  the  light  arrows.  The  reaction  to  this  up- 
ward thrust  exerted  upon  the  steam  by  the  movable  vanes  causes 
these  vanes  to  move  downward  (as  explained  in  connection 
with  Fig.  100,  Sec.  149,  and  as  indicated  by  the  heavy  arrows) 
with  an  enormous  velocity,  and  with  considerable  force. 

To  allow  for  the  expansion  of  the  steam,  the  above-mentioned 
tubular  shell  increases  in  diameter  to  the  right,  and  the  rotor 
vanes  increase  in  length  to  the  right.  The  stator  vanes  are  also 
longer  at  the  right. 

If  the  steam,  as  it  passes  to  the  right  from  the  turbine,  enters 
a  condenser,  the  effective  steam  pressure  and  likewise  the  ef- 
ficiency, will  be  increased  just  as  is  the  case  with  the  reciprocating 
^team  engine. 


STEAM  ENGINES  AND  GAS  ENGINES 


323 


236.  Carnot's  Cycle. — Nearly  a  century  ago,  the  French  physicist, 
Sadi  Carnot,  who  may  be  said  to  have  founded  the  science  of  thermody- 
namics, showed  by  a  line  of  reasoning  in  which  he  used  a  so-called  "ideal 
engine"  (Fig.  149),  that  by  taking  some  heat  HI,  from  one  body  and  giv- 
ing a  smaller  amount  H 2,  to  a  colder  body,  an  amount  of  heat  Hi—H2 
may  be  converted  into  work,  and  that  the  percentage  of  the  heat  that 
may  be  so  converted  depends  only  upon  the  temperatures  of  the  two 
bodies. 

In  Fig.  149  (Sketch  I-II),  let  a  cylinder  with  non-conducting  walls, 
a  non-conducting  piston,  and  a  perfect  conducting  base  in  contact  with 


7-77 


III-IV 


FIG.  149. 


the  perfect  conducting  "source"  S,  contain  some  gas  at  a  temperature  T\. 
(Parts  that  are  perfect  non-conductors  of  heat  are  shown  crosshatched.) 
Let  the  gas  be  a  perfect  gas,  i.e.,  one  which  obeys  Boyle's  law  and 
Charles's  law.  Let  7  be  a  perfectly  non-conducting  slab;  R,  the  perfectly 
conducting  "refrigerator,"  and  let  S  be  kept  constantly  at  the  tempera- 
ture TI,  and  R,  at  the  temperature  T2  on  the  absolute  scale. 

We  shall  now  put  the  gas  through  four  different  stages,  I,  II,  III,  and 
IV.  In  Fig.  149,  we  shall  indicate  the  four  processes  of  changing  from 
stages  I  to  II,  II  to  III,  III  to  IV,  and  from  IV  back  to  I,  by  the  four 
sketches  marked  respectively,  I-II,  II-III,  III-IV,  and  IV-I.  The  pis- 


324 


MECHANICS  AND  HEAT 


ton,  in  the  four  stages,  assumes  successively  the  positions  A,  B,  C,  and 
D,  and  the  corresponding  pressures  and  volumes  of  the  gas  are  indicated, 
respectively,  by  the  points  A,  B,  C,  and  D  on  the  pressure- volume 
diagram  (Fig.  150). 

Process  1:  As  the  gas  is  permitted  to  change  from  stage  I  to  II 
(sketch  marked  I-II)  by  pushing  the  piston  from  A  to  B,  it  does  work 
on  the  piston  (force  X  distance  or  pressure  X  volume,  Sec.  203),  and 
therefore  would  cool  itself  were  it  not  in  contact  with  the  perfect  con- 
ductor S.  This  contact  maintains  its  temperature  at  TV  A.,  i.e.,  the 
gas  takes  an  amount  of  heat,  say  HI,  from  source  S,  and  its  expansion  is 


L  Q        M  N 

Volume 

FIG.  150. 


represented  in  Fig.  150  by  the  portion  AB  of  an  isothermal.  Since  work 
is  the  product  of  the  average  pressure  and  the  change  in  volume  LM 
(Sec.  203),  we  see  that  the  work  done  by  the  gas  is  proportional  to,  and 
is  represented  by,  the  shaded  area  ABML. 

Process  2:  The  cylinder  is  next  placed  on  the  non-conducting  slab  7, 
and  the  gas  is  permitted  to  expand  and  push  the  piston  from  B  to  C.  In 
this  process  (sketch  II-III),  since  the  gas  is  now  completely  surrounded 
by  non-conductors  of  heat,  the  work  of  expansion  is  done  at  the 
expense  of  the  heat  of  the  gas  itself,  and  its  temperature  is  thereby  lowered. 
Consequently,  as  the  volume  increases,  the  pressure  decreases  more 
rapidly  than  for  the  previous  isothermal  expansion.  In  case  the  energy 
(heat)  of  expansion  must  come  from  the  gas  itself,  as  in  this  instance, 
the  expansion  is  Adiabatic.  AB  is  an  isothermal  line  and  BC  is  an  Adia- 
latic  line.  The  gas  is  now  at  stage  III,  and  the  work  done  by  the  gas 


STEAM  ENGINES  AND  GAS  ENGINES          325 

in  expanding  from  B  to  C  is  represented  by  the  area  BCNM  which  lies 
below  the  curve  BC  (compare  Process  1). 

Process  3:  The  cylinder  is  next  placed  upon  the  cold  body  or  "refrig- 
erator" R  (sketch  III-IV),  and  the  gas  is  compressed  from  C  to  D. 
Since  R  is  a  perfect  conductor,  this  will  be  an  isothermal  compression, 
and,  as  the  volume  is  slowly  reduced,  the  pressure  will  gradually  increase 
as  represented  by  the  isothermal  CD.  The  gas  is  now  at  stage  IV,  and 
is  represented  by  point  D  on  the  diagram.  The  work  done  upon  the 
gas  in  this  process  is,  by  previous  reasoning,  represented  by  the  area 
CDQN.  The  work  of  compressing  the  gas  develops  heat  in  it,  but  this 
heat,  say  HI,  is  immediately  given  to  the  refrigerator. 

Process  4:  Finally,  the  cylinder  is  again  placed  upon  the  non-conduct- 
ing slab  7,  and  the  piston  is  forced  from  D  back  to  the  original  position 
A.  Since  the  gas  is  now  surrounded  by  a  perfect  non-conductor,  the 
heat  of  compression  raises  its  temperature  to  T\.  As  the  volume  is 
gradually  decreased,  the  pressure  increases  more  rapidly  than  before,  be- 
cause of  the  accompanying  temperature  rise,  which  accounts  for  the  fact 
that  DA  is  steeper  than  CD.  In  this  case,  of  course,  we  have  an  Adia- 
batic  Compression  and  the  line  DA  is  an  adiabatic  line.  The  work 
done  upon  the  gas  in  this  process  is  represented  by  the  area  DALQ. 

Efficiency  of  Carnot's  Cycle. — From  the  preceding  discussion, 
we  see  that  the  work  done  by  the  gas  during  the  two  expansions 
(Processes  1  and  2)  is  represented  by  the  area  below  ABC; 
while  the  work  done  upon  the  gas  during  the  two  compressions 
(Processes  3  and  4)  is  represented  by  the  area  below  ADC. 
Consequently,  the  net  work  obtained  from  the  gas  is  represented 
by  the  area  A  BCD. 

It  has  just  been  shown  (Process  1)  that  the  gas  as  it  ex- 
pands from  A  to  B,  does  work  represented  by  the  area  ABML, 
and  since  its  temperature  remains  constant,  it  must  take  from 
the  source  S,  an  amount  of  heat  energy  equal  to  this  work. 
Let  us  call  this  heat  Hi.  Similar  reasoning  shows  that  when  com- 
pressed from  C  to  D,  the  gas  gives  to  the  refrigerator  an  amount 
of  heat  Hz  represented  by  the  area  CDQN.  During  the  other  two 
processes  (adiabatic  processes)  the  gas  can  neither  acquire  nor 
impart  heat.  Accordingly,  for  this  cycle,  the  efficiency  is  given 
by  the  equation 

F_   work  done    _  ABC  D  _  ABML— CDQN  (  >,_#!— #? 

'  "heat  received  ~  ABML  "         ABML  Hl 

(88) 

Now,  the  heat  contained  by  a  gas,  or  any  other  substance,  is 


326  MECHANICS  AND  HEAT 

proportional  to  the  temperature  of  the  substance  (assuming  that 
the  body  has  a  constant  specific  heat).     Consequently, 


This  equation  shows  (as  mentioned  in  Sec.  228)  that  if  the  ab- 
solute temperature  TI  of  the  "live"  steam  as  it  enters  the 
cylinder  from  the  boiler  is  400°  A.  and  the  temperature  T2  of 
the  condenser  is  300°  A.,  then  the  maximum  theoretical  efficiency 

.       .    400-300      oe 
of  the  engine  is  —  TQX  —  or  25  per   cent.     For  a  rigorous,  and 

more  extended  treatment  of  this  topic  consult  advanced  works. 

237.  The  Gas  Engine  —  Fuel,  Carburetor,  Ignition,  and  Gover- 
nor. —  In  the  gas  engine,  the  pressure  which  forces  the  piston  along 
the  cylinder  is  exerted  by  a  hot  gas,  instead  of  by  steam  as  in  the 
case  of  the  steam  engine.  The  gas  engine  also  differs  from  the 
steam  engine  in  that  the  fuel,  commonly  an  explosive  mixture 
of  gasoline  vapor  with  air,  is  burned  (i.e.,  explosion  occurs) 
within  the  cylinder  itself.  For  this  reason,  no  furnace  or  boiler 
is  required,  which  makes  it  much  better  than  the  steam  engine 
for  a  portable  source  of  power.  Gas  engines  may  be  made  very 
light  in  proportion  to  the  power  which  they  will  develop.  The 
weight  per  H.P.  varies  from  several  hundred  pounds  for  station- 
ary engines,  to  10  Ibs.  for  automobiles.  As  has  already  been 
mentioned,  the  lightness  of  the  gas  engine  (as  low  as  2.5  Ibs. 
per  H.P.  for  aeroplanes)  has  made  aeroplane  flight  possible. 

The  fact  that  a  gas  engine  may  be  started  in  an  instant  (i.e., 
usually),  and  that  the  instant  it  is  stopped  the  consumption  of 
fuel  ceases,  makes  it  especially  adapted  for  power  for  automo- 
biles, or  for  any  work  requiring  intermittent  power.  The  fact 
that  the  power  can  be  instantly  varied  as  required  is  also  a  point 
in  its  favor. 

Fuel.  —  Gasoline  is  the  most  widely  used  fuel  for  gas  engines. 
It  is  readily  vaporized,  and  this  vapor,  mixed  with  the  proper 
amount  of  air  as  it  is  drawn  into  the  cylinder,  is  very  explosive 
and  is  therefore  readily  ignited.  Complete  combustion  is 
easily  obtained  with  gasoline;  so  that  it  does  not  foul  the  cylinder 
as  some  fuels  do.  Kerosene  is  much  less  volatile  than  gasoline, 
but  may  be  used  after  the  cylinder  has  first  become  heated  by  the 
use  of  gasoline.  Alcohol  may  also  be  used.  Crude  Petroleum 
is  used  in  some  engines.  Illuminating  Gas,  mixed  with  air,  may 


STEAM  ENGINES  AND  GAS  ENGINES          327 

be  used  as  a  fuel.  Natural  gas,  where  available,  forms  an  ideal 
fuel,  and  is  used  in  some  large  power  plants.  The  use  of  "Pro- 
ducer" Gas  requires  considerable  auxiliary  apparatus,  but  be- 
cause of  its  cheapness,  it  is  profitably  used  by  stationary  engines. 

Briefly,  producer  gas  is  formed  by  heating  coal  while  re- 
stricting the  air  supply,  so  that  the  carbon  burns  to  carbon 
monoxide  (CO)  which  is  a  combustible  gas,  instead  of  to  carbon 
dioxide  (CO2),  which  is  incombustible.  If  some  steam  (H2O) 
is  admitted  with  the  air,  the  steam  is  decomposed  into  oxygen 
(O)  which  combines  with  the  carbon  and  forms  more  carbon 
monoxide  (CO).  The  remaining  hydrogen  constituent  (H)  of 
the  steam  is  an  excellent  fuel  gas.  All  of  these  gases  pass  from 
the  coal  through  various  cooling  and  purifying  chambers,  either 
directly  into  the  gas  engine,  or  into  a  gas  tank  to  be  used  as 
required. 

The  Carburetor. — The  carburetor  is  a  device  for  mixing  the 
vapor  of  the  gasoline,  or  other  liquid  fuel,  with  the  air  which 
passes  into  the  cylinder,  thus  forming  the  "charge."  The  ex- 
plosion of  this  charge  develops  the  pressure  which  drives  the 
piston.  As  the  air  being  drawn  into  the  engine  rushes  past  a 
small  nozzle  connected  with  the  gasoline  supply  (see  C,  Fig. 
153,  left  sketch),  the  gasoline  is  "drawn"  out  of  the  nozzle  (see 
atomizer,  Sec.  156)  in  the  form  of  a  fine  spray,  which  quickly 
changes  to  vapor,  and  is  thereby  thoroughly  mixed  with  the  air 
to  form  the  "charge."  This  thorough  mixing  is  essential  to 
complete  combustion.  If  kerosene  is  used,  the  air  must  be  pre- 
viously heated  in  order  to  vaporize  the  spray.  It  is  well  to  pre- 
heat the  air  in  any  case. 

Ignition. — The  charge  is  usually  ignited  electrically,  either 
by  what  is  called  the  "jump  spark"  from  an  induction  coil,  or 
by  the  "  make-and-break "  method.  An  induction  coil  consists 
of  a  bundle  of  iron  wires,  upon  which  is  wound  a  layer  or  two  of 
insulated  copper  wire,  called  the  primary  coil.  One  end  of  this 
primary  coil  is  connected  by  a  wire  directly  to  one  terminal  of 
a  battery,  while  the  other  end  is  connected  to  the  opposite  ter- 
minal of  the  battery  through  a  vibrator  or  other  device,  which 
opens  and  closes  the  electrical  circuit  a  great  many  times  per 
second.  On  top  of  the  primary  coil,  and,  as  a  rule,  carefully  in- 
sulated from  it,  are  wound  a  great  many  turns  of  fine  wire,  called 
the  secondary  coil.  When  the  current  in  the  primary  circuit  is 
broken,  a  spark  will  pass  between  the  terminals  of  the  second- 


328  MECHANICS  AND  HEAT 

ary,  provided  they  are  not  too  far  apart.  The  "spark  distance" 
of  the  secondary  varies  from  a  small  fraction  of  an  inch  to  several 
feet,  depending  upon  the  size  and  kind  of  induction  coil.  For 
ignition  purposes,  only  a  short  spark  is  required.  By  means  of  a 
suitable  mechanism,  this  spark  is  made  to  take  place  between 
two  points  in  the  " spark  plug"  (B,  Fig.  153)  within  the  cylinder 
at  the  instant  the  explosion  should  occur. 

In  the  "make-and-break"  method  of  ignition,  neither  the 
secondary  nor  the  vibrator  is  needed.  One  terminal  of  the 
primary  coil,  which,  with  its  iron  wire  "core,"  is  called  a  "spark 
coil,"  is  connected  directly  to  the  firing  pin  which  passes  through 
a  hole  into  the  cylinder.  The  other  terminal  of  the  primary  is 
connected  to  one  pole  of  a  battery.  From  the  other  pole  of  the 
battery  a  wire  leads  to  a  metal  contact  piece  which  passes  into 
the  cylinder  from  which  it  is  insulated,  at  a  point  near  the  firing 
pin.  By  means  of  a  cam,  this  firing  pin  is  made  to  alternately 
touch  and  then  move  away  from  the  metal  contact  piece  within 
the  cylinder.  Consequently,  by  proper  adjustment  of  the  cam, 
the  circuit  is  broken  by  the  firing  pin  and  the  gas  is  ignited  at 
the  instant  the  explosion  is  desired.  If  the  spark  occurs  when 
the  piston  is  past  dead  center  it  is  said  to  be  retarded,  if  before, 
advanced.  Engines  running  at  very  high  speed  require  the  spark 
to  be  advanced,  or  the  flame  will  not  have  time  to  reach  all  of 
the  gas  until  rather  late  in  the  stroke.  The  indicator  card  will 
tell  whether  or  not  advancing  the  spark  increases  the  power  in 
a  given  instance.  If  the  spark  is  advanced  too  far  "back-firing" 
results,  with  its  attendant  jarring  and  reduction  of  power. 

The  electric  current  may  be  produced  by  a  "magneto."  The 
magneto  generates  current  only  when  the  engine  is  running;  so 
that  a  battery  must  be  used  when  starting  the  engine,  after  which, 
by  turning  a  switch,  the  magneto  is  thrown  into  the  circuit  and 
the  battery  is  thrown  out. 

Cooling. — To  prevent  the  cylinder  from  becoming  too  hot,  a 
"water  jacket"  is  provided.  The  cylinder  walls  are  made 
double,  and  the  space  between  them  is  filled  with  water.  This 
water,  as  it  is  heated,  passes  to  the  "radiator"  and  then  returns 
to  the  water  jacket  again.  The  water  circulation  is  maintained 
either  by  a  pump,  or  by  convection.  The  radiator  is  so  con- 
structed, that  it  has  a  large  radiating  surface.  A  fan  is  frequently 
used  to  cause  air  to  circulate  through  the  radiator  more  rapidly. 
In  some  automobiles  air  cooling  is  used  entirely,  the  cylinder 


STEAM  ENGINES  AND  GAS  ENGINES          329 

being  deeply  ribbed  so  as  to  have  a  large  surface  over  which  the 
air  is  forced  in  a  rapid  stream. 

The  Governor. — Commonly  some  form  of  the  Centrifugal  Gover- 
nor (Sec.  63)  is  used  to  control  the  speed.  In  the  "  hit-or-miss  " 
method  no  "charge"  is  admitted  when  the  speed  is  too  high. 
This  causes  fluctuations  in  the  speed  which  are  readily  noticeable. 
In  other  methods  of  speed  control,  either  the  quantity  of  "rich- 
ness" (proportion  of  gas  or  gasoline  vapor  to  air)  of  the  charge 
is  varied  to  suit  the  load.  If  the  load  is  light,  the  governor  re- 
duces the  gas  or  gasoline  supply;  or  else  it  closes  the  intake 
valve  earlier  in  the  stroke,  thereby  reducing  the  quantity  of  the 
charge. 

238.  Multiple-cylinder    Engines. — With    two-cycle    engines 
(Sec.  240),  an  explosion  occurs  every  other  stroke;  while  in  the 
four-cycle  engine  (Sec.  239)  explosions  occur  only  once  in  four 
strokes  (i.e.,  in  two  revolutions).     It  will  be  seen  that  the  applied 
torque  is  quite  intermittent  as  compared  with  that  of  the  steam 
engine.     If  an  engine  has  six   cylinders,  with  their  connecting 
rods  attached  to  six  different  cranks  on  the  same  crank-shaft, 
then,  by  having  the  cranks  set  at  the  proper  angle  apart  and  by 
properly  timing  the  six  different  explosions,  a  nearly  uniform 
torque  is  developed.     The  six-cylinder  engine  is  characterized 
by  very  smooth  running.     The  four-cylinder  engines,  and  even 
the  two-cylinder  engines,  produce  a  much  more  uniform  torque 
than  the  single-cylinder  engines. 

239.  The  Four-cycle  Engine. — In  the  so-called  four-cycle  en- 
gine, a  complete  cycle  consists  of  four  strokes,  or  two  revolu- 
tions.    The  four  strokes  are,  suction  or  charging,  compression, 
working,   and  exhaust.     The  stroke,  at  the  beginning  of  which 
the  explosion  occurs,  is  the  working  stroke.     With  this  engine, 
every  fourth  stroke  is  a  working  stroke;  whereas,  in  the  steam 
engine,  every  stroke  is  a  working  stroke.     The  operation  of  this 
engine  will  be  understood  from  a  discussion  of  Fig.   151.     In 
the  upper  sketch,  marked  I  (Fig.  151),  valve  a  is  open  and  valve 
6  is  closed,  so  that  as  the  piston  moves  to  the  right  the  "suction" 
draws  in  the  charge  from  the  carburetor.     This  is  the  charging 
stroke.     On  the  return  stroke  of  the  piston  (Sketch  II),  both 
valves  are  closed  and  the  charge  is  highly  compressed. 

As  the  piston  reaches  the  end  of  its  stroke,  the  gas  then  oc- 
cupying the  clearance  space,  or  "combustion  chamber,"  is  ignited 
by  means  of  either  the  "  firing  pin  "  c  or  a  "  spark  plug,"  depending 


330 


MECHANICS  AND  HEAT 


upon  which  method  of  ignition  is  used.  Ignition  may  occur 
either  at,  before,  or  after  "dead  center."  (See  Ignition,  Sec. 
237.)  The  "explosion,"  or  the  burning  of  the  gasoline  vapor, 
produces  a  very  high  temperature  and  therefore,  according  to 


FIG.  151. 


FIG.  152. 

the  law  of  Charles,  a  very  high  pressure.  This  high  pressure 
pushes  the  piston  to  the  right.  This  stroke  is  called  the  working 
stroke  (Sketch  III).  As  the  piston  again  returns  to  the  left, 
valve  b  is  open,  and  the  burned  gases  escape.  This  is  the  exhaust 


STEAM  ENGINES  AND  GAS  ENGINES 


331 


stroke.  The  exhaust  is  very  noisy  unless  the  exhaust  gases  are 
passed  through  a  muffler. 

In  many  engines  there  is  no  piston  rod,  the  connecting  rod 
being  attached  directly  to  the  piston  as  shown.  The  valves 
are  operated  automatically  by  cams,  or  other  devices  connected 
with  the  crank  shaft  so  that  by  proper  adjustment,  exact  timing 
may  be  obtained. 

Indicator  Card. — An  indicator  mechanism  may  be  connected 
with  the  cylinder  just  as  with  the  steam-engine  cylinder  (Sec.  234). 
In  Fig.  152,  is  shown  the  indicator  card  from  a  four-cycle  engine. 
The  line  marked  1  shows  the  pressure  corresponding  to  the 
charging  stroke  (Sketch  I,  Fig.  151).  Line  2  shows  the  pressure 
during  the  compression  stroke  (Sketch  II,  Fig.  151).  At  point 


FIG.  153. 

e,  the  explosion  has  occurred,  and  the  pressure  has  reached  a 
maximum.  Line  3  represents  the  pressure  during  the  working 
stroke  (Sketch  III),  showing  how  it  varies  from  the  maximum 
down  to  /.  Line  4  shows  the  pressure  during  the  exhaust  stroke 
(Sketch  IV,  Fig.  151). 

240.  The  Two-cycle  Engine. — The  operation  of  the  two-cycle 
engine  will  be  understood  from  a  discussion  of  Fig.  153.  As 
the  piston  moves  upward,  compressing  a  previous  charge,  it 
produces  suction  at  port  a  (left  sketch),  and  draws  in  the  charging 
gas  from  the  carburetor  C  into  the  crank  case  A,  which  is  air- 
tight in  this  type  of  engine.  As  the  piston  reaches  the  top  of  its 


332  MECHANICS  AND  HEAT 

stroke,  the  charge  is  ignited  by  the  spark  plug  B,  and  explosion 
occurs.  As  the  piston  now  descends  it  is  driven,  with  great  force, 
by  the  high  pressure  of  the  heated  gases.  This  is  the  working 
stroke.  As  soon  as  the  piston  passes  below  the  exhaust  port 
6  (right  sketch)  the  exhaust  gas  escapes,  in  part.  An  instant 
later,  the  piston  is  below  port  c,  and  part  of  the  gas  in  the  crank 
case,  which  gas  is  now  slightly  compressed  by  the  descent  of 
the  piston,  rushes  through  port  c.  As  this  charge  enters  port 
c,  it  strikes  the  baffling  plate  D,  which  deflects  it  upward,  thus 
forcing  most  of  the  remaining  exhaust  gas  out  through  port  6. 
As  the  piston  again  rises,  it  compresses  this  charge  preparatory 
to  ignition,  and  the  cycle  is  completed. 

PROBLEMS 

1.  If  all  of  the  energy  developed  by  a  mass  of  iron  in  falling  778  ft.  is 
used  in  heating  it,  what  will  be  its  temperature  rise? 

2.  If  the  complete  combustion  of  1  Ib.  of  a  certain  grade  of  coal  develops 
13,000  B.T.U.'s  of  heat,  how  much  work  (in  ft.-lbs.)  would  it  perform  if 
it  is  used  in  a  heat  engine  of  10  per  cent,  efficiency? 

3.  How  many  H.P.-hours  of  potential  energy  does  a  pound  of  coal 
(13,500  B.T.U.'s  per  Ib.)  possess,  and  how  many  H.P.-hours  of  work  can  a 
good  steam  engine  (say  of  12.5  per  cent,  efficiency)  obtain  from  it?     Note 
that  one  horse-power  for  one  second  is  550  ft.-lbs. 

4.  How  long  would  a  ton  of  coal,  like  that  mentioned  in  Problem  2,  run  a 
10-H.P.  steam  engine  of  6  per  cent,  total  efficiency? 

6.  How  high  would  the  heat  energy  (14,000  B.T.U.'s  per  Ib.)  from  a  given 
mass  of  coal  lift  an  equal  mass  of  material,  if  it  were  possible  to  convert  all 
of  the  heat  of  the  coal  into  mechanical  energy? 

6.  Find  the   H.P.  of  a  noncondensing  steam  engine  supplied  during  full 
stroke  with  steam  at  80  Ibs.  per  sq.  in.  pressure  (80  Ibs.  is  the  available 
working  pressure),  when  making  120  R.P.M.  (4  strokes  per  sec.) ;  the  length 
of  stroke  being  2  feet  and  the  cross  section  of  the  piston  being  150  sq.  in. 

7.  How  many  pounds  of  water  at  70°  F.  will  be  changed  to  steam  at  212° 
F.  for  each  pound  of  coal  (Prob.  2)  burned  in  a  furnace  of   90  per  cent, 
efficiency,  heating  a  boiler  of  70  per  cent,  efficiency. 

8.  An  engine  whose  speed  is  150  R.P.M.,  has  a  piston  15  in.  in  diameter 
which  makes  a  2-ft.  stroke.     The  indicator  diagram  is  4  in.  long  and  has  an 
area  of  9  sq.  in.     The  indicator  spring  is  a  ''50-lb.  spring,"  i.e.,  a  rise  of 
1  in.  by  the  indicator  pencil  indicates  a  change  in  pressure  of  50  Ibs.  per  sq. 
in.     What  is  the  power  of  the  engine? 

9.  Find  the.  H.  P.  of  the  engine  (Prob.  6)  with  cut-off  set  at  one-quarter 
stroke,  the  average  pressure  during  the  remaining  3/4  stroke  being  30  Ibs. 
per  sq.  in. 

10.  Find  the  H.P.  of  the  engine  (Prob.  6)  with  cut-off  at  half  stroke,  the 
pressure  during  the  last  half  of  the  stroke  being  30  Ibs.  per  sq.  in. 


STEAM  ENGINES  AND  GAS  ENGINES          333 

11.  How  many  B.T.U.'s  will  a  1/2-oz.  bullet  develop  as  it  strikes  the  target 
with  a  velocity  of  1800  ft.  per  sec.?     If  this  heat  were  all  absorbed  by  the 
bullet  (lead)  what  would  be  its  temperature  rise? 

12.  What  is  the  limiting  theoretical  efficiency  (thermodynamic  efficiency) 
of  a  steam  engine  whose  boiler  is  at  180°  C.,  and  whose  condenser  is  at 
50°  C.? 


INDEX 


The  numbers  refer  to  pages. 


Absolute  temperature  scale,  237 

zero,  236 

Absorption  of  heat,  297 
Accelerated  motion,  uniform,  26,  28 
Accelerating  force,  26,  49,  50,  51 
in  circular  motion,  72 
in  free  fall,  35 
in  simple  harmonic  motion, 

83,84 
Accelerating  torque,  66,  68 

equation  for,  67 
Acceleration,  angular,  62 

with  Atwood's  machine,  41 
of  gravity,  35 

variation  of,  35 
linear,  25,  29 

and  angular  compared,  63 
radial,  73 
in  simple  harmonic  motion,  83, 

84 

uniform  and  nonuniform,  26,  29 
Action  and  reaction,  49 
applications  of,  51 
Actual  mechanical  advantage,  111 
Addition  of  vectors,  12 
Adhesion  and  cohesion,  141 
fish  glue  for  glass,  142 
Adiabatic,    compression  and  expan- 
sion, 324 
line,  324 

and  isothermal  processes,  324 
Air  compressor,  201 
Air  friction,  on  air,  177,  178 

effect  on  falling  bodies,  36 
on  meteors,  181 
on  projectiles,  46 
Air,  liquefied  and  frozen,  278 
liquid,  278,  279,  280 

properties   and    effects   of, 
281 


Air  pump,  mechanical,  200 

mercury,  201 
Alloys,  melting  point,  255 
Altitude  by  barometer,  187 
Amalgams,  156 
Ammonia,  156 

refrigerating  apparatus,  272 
Amplitude,  87,  293 
Andrews,  work  on  critical  tempera- 
ture, 273 

isothermals  of  carbon  dioxide, 

274 

Aneroid  barometer,  186 
Angle  of  elevation,  47 

of  shear,  152 

unit  of,  62 

Angular  acceleration  and  velocity, 
62 

and  linear  velocity  and  accel- 
eration compared,  63 

measurement,  62 

velocity,  average,  63 
Antiresultant  force,  16 
Aqueous  vapor,  pressure  of,  262,  304 
Archimedes'  principle,  163 

application  to  gases,  182 

and  floating  bodies,  165 

experimental  proof  of,  164 
Army  rifle,   range  and  velocity  of 

projectile,  46 
Artificial  ice,  272 
Aspirator,  or  filter  pump,  209 
Atmosphere,  composition  of,  180 

height  of,  181 

moisture  of,  302 

pressure  of,  183,  184,  197,  199 

standard,  185 
Atomic   heat,    Dulong   and    Petit's 

law,  246 
Atomizer,  209 


335 


336 


INDEX 


Attraction,  gravitational,  30 
Atwood's  machine,  41 
Avogadro's  law,  180 
Axis  of  rotation,  23 

Balance  wheel,  of  watch,  tempera- 
ture compensation  of,  233 
Balanced  columns,  density  by,  162 

forces,  so-called,  51 
Ball  and  jet,  212 

bearings,  102 
Ballast,  use  and  placing  of,  in  ships, 

127 
Ballistic  pendulum,  and  velocity  of 

rifle  bullet,  55 

Balloon,  lifting  capacity  of,  183 
Barometer,  aneroid,  186 

mercury,  184 

uses  of,  187 
Barometric  height,  185 
Baseball,  curving  of,  213 
Beam  balance,  127,  128,  129 
Beams,     horizontal,    strength    and 

stiffness  of,  150 
Bearings,  ball,  102 

roller,  103 

babbitt  in,  101 
Beats,  in  sound,  293 
Belt  speed  and  angular  speed,  64 
Bernoulli's  theorem,  209,  210,  211 
Black  body  radiation,  297 
Block  and  tackle,  115 
Blood,  purification  of,  158 
Blowers,  rotary,  203 
Boiler  explosions  and  superheating, 
266 

"scale,"  287 
Boiling,  261 
Boiling  point,  defined,  262 

at  high  altitudes,  264 

effect  of  dissolved  substance  on, 
262 

effect  of  pressure  on,  262 

tables  of,  262 
Boyle's  law,  179,  187,  192,  317 

deviation  from,  277 

and  kinetic  theory,  188 
Brake,  Prony,  106,  107 
Breaking  stress,  149 


British  system  of  units,  2 

thermal  units,  or  B.T.U.,  218, 

243,  311 
Brittleness,  144 
Brownian  motion,  139 
Bulk  or  volume  modulus,  152 
Bullet,  determination  of  velocity,  55 

velocity  at  different  ranges,  46 
'"Bumping,"  due  to  superheating  of 

water,  265 
Buoyancy,  center  of,  166 

of  gases,  182 

of  liquids,  162 
Buoyant  force,  162 

Cailletet,  liquefaction  of  gases,  278 
Calibration  of  thermometer,  223 
Caliper,  micrometer,  7 

vernier,  5 

Calms,  zone  of,  305 
Caloric  theory  of  heat,  218 
Calorie,  243 
Calorimeter,  Bunsen's  ice,  251 

Joly's  steam,  252 

water  equivalent  of,  244 
Calorimetry,  243 

Camphor,  effect  of,  on  surface  ten- 
sion, 173 
Canal    boat,    discussion    of    inertia 

force,  51 

Cannon,    "shrinking"   in   construc- 
tion of,  228 

Capacity,  thermal,  244 
Capillarity,  173 

Capillary  rise,  in  tubes,  wicks  and 
soils,  174 

tubes,  174 

Car  and  hoop  on  incline,  98 
Carbon  dioxide,  cooling  effect  of,  270 

isothermals  of,  274,  276 

"snow,"  271 
Carburetor,  327 

Card  and  spool  experiment,  213 
Carnot,  Sadi,  French  physicist,  323 

cycle,  323,  324 

efficiency  of,  325 

Carnot's  "ideal"  engine,  313,  323 
Cascade  method  of  liquefying  gases, 
279 


INDEX 


337 


Castings,  when  clear-cut,  256 
Cavendish,  gravitational  experiment 

of,  30 

Center  of  buoyancy,  166 
of  gravity,  122 

effect  on  levers,  123 
of  mass,  124 
of  population,  124 
Centigrade  scale  of  temperature,  224 
Centimeter,  denned,  4 
Centimeter-gram-second  (C.  G.  S.) 

system,  4 
Central  force,  72 

radial,  75 

Centrifugal  blowers,  203 
cream  separator,  76 
dryer,  73 
force,  72 

effect  on  shape  of  earth,  73 
practical  applications  of,  73, 

76,  77,  79 
governor,  79,  315 
pump,  204 
Centripetal  force,  72 
Chain  hoist,  121 
Change  of  state,  219,  220,  250 
Charles'  law,  236 
Chemical  hygrometer,  the,  303 
Choke  damp,  181 
Circular  motion,  acceleration  radial 

in,  75 

uniform,  72 

Circulation  of  air  due  to  stove,  284 
Clepsydra,  10 
Clinical  thermometer,  225 
Clock,  essentials  of,  9 
Clouds,  height,  character  and  veloc- 
ity, 302 
Coefficient  of  cubical  expansion,  234 

table,  235 
of  friction,  101 

determination  of,  101 
limits  maximum  pull  of  loco- 
motive, 102 
of  linear  expansion,  229 

differences    in,    and    applica- 
tions of,  230-234 
table  of,  230 
Cohesion,  141 


Cold  denned,  219 

produced  by  evaporation,  268 
by  expansion  of  gas,  246,  278, 

280 
Combustion,  defined,  248 

heats  of,  table,  249 
Compensated  balance  wheel,  233 

pendulum,  234 
Components  of  forces  and  velocities, 

19,  20 
Compressibility  of  gases,  178,  179 

of  water,  155,  165 
Compressor,  air,  201 
Compound  lever,  130 
Condenser,  jet,  316 

surface,  316 

Condensing  steam  engine,  316 
Conditions  of  equilibrium,  the  two, 

64 

Conduction  of  heat,  286 
Conductivity,  thermal,  288 

table,  289 

Cone,  equilibrium  of,  126 
Conservation  of  energy,  93,  210,  251 

of  mass,  139 

of  matter,  139 

momentum,  three  proofs  of,  53, 

54 

Convection,  283,  285 
Conversion  of  units,  4 
Cooling  effect  of  evaporation,  268, 

270 

of  internal  work,  gases,  278 
Cooling,  Newton's  law  of,  297 

Stefan's  law  of,  297 
Cornsheller,  fly  wheel  on,  69 
Couple,  the,  61 
Crane,  the,  17 
Cream  separator,  the,  76 
Crew,  Henry.     See  Preface. 
Critical    temperature,    and    critical 
pressure,  273 

simple  method  of  determining, 
277 

table  of,  274 

Cubical  expansion,  coefficient  of,  234 
Curves,  plotting  of  and  use,  48 

elevation  of  outer  rail  at,  77 
Curving  of  baseball,  213 


338 


INDEX 


Cut-off  point,  steam  engine,  316 
controlled  by  governor,  315 

Cyclones,  306 

cause  of  rotary  motion,  307 

d'Alembert,  principle  of,  49,  51 

Davy's  safety  lamp,  287 

Day,  the  siderial  and  mean  solar,  3 

"Dead  air"  space  in  buildings,  287 

Density,  defined,  139 
of  earth,  average,  30 
of  liquids  by  balanced  columns, 

161 

of  solids,  liquids  and  gases,  140 
not  specific  gravity,  166 
of  some  substances,  table  of,  140 
of  water,  maximum,  255 

Deserts,  cause  of,  305 

Dew,  302 

point,  303 
and  frost,  303 

Dewar  flask  or  thermal  bottle,  282 
liquefaction  of  gases,  278 

Dialysis,  158 

Differential  pulley,  121 
Wheel  and  axle,  122 

Diffusion  of  gases,  178,  180 
of  liquids,  156 

Diminution  of  pressure  in  regions  of 
high  velocity,  208 

Disc  fan,  203 

Displacement,   in  simple   harmonic 

motion,  84 
of  a  ship,  165 

Dissipation  of  energy,  99 

Distance,  fallen  in  a  given  time,  40 
law  of  inverse  squares  of,  31 
either  scalar  or  vector,  24 
traversed  in  a  given  time,  41 

Drains,  flow  in,  196 

Driving  inertia  force,  51 

work  of,  90 
torque,  69 

Ductility,  144 

Dufour,  superheating  of  water,  256 

Dulong  and  Petit's  law,  246 

Dynamometers,      absorption      and 
transmission,  106 

Dyne,  the,  27,  36 


Earth,  atmosphere  of,  180 

attraction  on  the  moon,  33 

average  density  of,  30 

path  of,  3,  4,  32,  34 

weight  of,  30 

Earth's  rotation,  effect  on  shape  of, 
73 

effect  on  moving  train,  307 

and  trade  winds,  305 
Ebullition  and  evaporation,  260 
Eccentric,  the,  318 
Effects  of  heat,  219 
Efficiency  of  Carnot's  cycle,  325 

of  cream  separator,  76 

of  gas  engine,  313 

of  simple  machines,  111,  112 

of    steam    engine,    boiler,   and 
furnace,  312 

of  steam  engine,  calculation  of, 

313 

Efflux,  velocity  of,  196 
Elastic  fatigue,  145,  149 

limit,  145 

rebound,  explanation  of,  145 
Elasticity,  general  discussion  of,  142 

of  gases,  178 

of  shearing,  or  of  torsion,  151, 
152 

of  tension  or  of  elongation,  146 

of  volume   or  of  compression, 
151,  152 

perfect,  142 

three  kinds  of,  151 
Electric  fan  and  windmill,  202 

fire  alarm,  231 
Electrical  effect  of  heat,  242 
Elements  and  compounds,  138 
Elevation  of  outer  rail  on  a  curve,  77 
Elevator,  hydraulic,  206 
Energy,  chemical,  218 

conservation  of,  93,  94 

defined,  92 

dissipation  of,  99 

heat,  a  form  of,  217,  243,  244 

kinetic,  92,  96 

potential,  92,  95 

of  a  rotating  body,  96,  97,  98 

of  sun,  218 

sources  of,  218 


INDEX 


339 


Energy,  transformation  of,  93,  94 

transformation  of  involves  work, 
93,  94 

units  of,  95 
Engineer's  units  of  mass  and  force, 

37 

Equilibrant,  16 

Equilibrium  of  rigid  body,  two  con- 
ditions of,  64 

on  inclined  plane,  126 

of  rocking  chair,  126 

in  vaporization,  266 

of  wagon  on  hillside,  127 

stable,   unstable,   and   neutral, 

126 

Erg,  90 
Ether,  the,  295 

waves  in,  291 
Evaporation,  cooling  effect  of,  268 

and  ebullition,  260 
Evener,  two-horse,  129 
Expansibility  of  gases,  177,  179 
Expansion,    apparent,    of  mercury, 
223 

of  solids,  230 

and  temperature  rise,  221 
Expansive  use  of  steam,  316 

Factor  of  safety,  149 

Fahrenheit's  thermometric  scale,  224 

Falling  bodies,  laws  of,  38-48 

maximum  velocity  in  air,  36 
Fan,  two  kinds,  203 
Faraday,    Michael,    liquefaction    of 

gases,  278 
''Film,"  width  of,  172,  173 

work  in  forming,  172 
Fire  alarm,  electric,  231 

damp,  181 

syringe,  311 

Fish  glue,  adhesion  to  glass,  142 
Fleuss  or  Geryk  pump,  201 
Flight  of  aeroplane,  52 

of  birds  when  starting,  52 
Floating  bodies,  165 

immersed,  164 

Flow  of  liquids,  gravitational,  196 
Fluids,  in  motion,  properties  of,  194 
Flux,  use  of,  141 


Flywheel,  bursting  of,  75 

calculation  of,  98 

design,  98 

kinetic  energy  of,  97 

speed  regulation  by,  98 

use  of,  97 

Foot-pound  and  foot-poundal,  90 
Force,  accelerating,  26,  49,  50 

"arm,"  levers,  114 

buoyant,  162 

central,  72 

centrifugal,  72 

centripetal,  72 

defined,  26 

driving  inertia,  51 

impulsive,  52 

resisting,  110 

resolution  of  into  components 
19,  101 

of  restitution  in  simple  har- 
monic motion,  83,  84,  85, 
86 

units  of,  27,  36 

working,  110 

Forced  draft,  locomotive,  209 
Forces,  addition  of,  11, 16 

balanced,  16,  51 

graphical  representation  of,  12 

in  planetary  motion,  32 

polygon  of,  16     , 

resolution  of,  19 

torque  due  to,  60 
Four-cycle  gas  engine,  329 
Franklin,  Benjamin,  experiment  on 

boiling  point,  263,  264 
Freezing  mixtures,  258,  259 

point  of  solutions,  255 

lowering   of  by  pressure,  256 
Friction,  cause  of,  100 

beneficial  effects  of,  101 

coefficient  of,  101 

head,  177,  194,  196 

internal,  100 

kinetic,  100 

laws  of,  100 

of  air  on  projectiles,  46 

rolling,  102,  103 

sliding,  99 

static,  101 


340 


INDEX 


Friction,  useful,  101 

work  of,  90,  103,  104 

produces  heat,  99 
Fulcrum,  113 
Fundamental  quantities,  1 

units,  2 

Furnace,  efficiency  of,  312 
Fusion  of  alloys,  255 

and  change  in  volume,  256 

heat  of,  250 

and  melting  point,  255 

Gas,  general  law,  240 

laws,  summary  of  three,  239 

thermometer,  226 
Gases,  compressibility  of,  179 
table  of  densities  of,  140 

diffusion  of,  178,  180 

general  law  of,  240 

kinetic  theory  of,  179,  236 

two  specific  heats  of,  246 

thermal  conductivity  of,  289 

and  vapors,  distinction  between, 
277 

average  velocity  of  molecules, 

180 
Gas  engine,  326 

carburetor,  327 

combustion  chamber,  329 

efficiency  of,  313 

four-cycle,  329 

fuel,  326 

governor,  329 

ignition,  327 

indicator  card  of,  331 

"make-and-break"  ignition,  328 

multiple  cylinder,  329 

"richness"  of  charge,  329 

six-cylinder,  329 

spark  plug,  328 

two-cycle,  331 

very  light,  for  aeroplanes,  313 

water  jacket,  328 

Gelatine  film,  adhesion  to  glass,  142 
Geryk  or  Fleuss  pump,  201 
Geysers,  265 

artificial,  266 
Glaciers,  explanation  of  motion,  257 

location  of,  258 


Glaciers,  origin  of,  258 

velocity  of,  258 
Gold  filling  of  teeth,  141 

foil,  144 

Governor,  the  centrifugal,  79,  315 
Gram  mass,  defined,  4 

weight,  defined,  36 
Graphical  method  and  vectors,  12 

representation,  of  space  passed 
over  by  a  falling  body,  39 
of  force,  16 
of  velocity,  12,  13 
Gravitation,  Newton's  laws  of,  30 

units  of  energy,  why  chosen,  95 

universal,  30 
Gravity,  acceleration  of,  35 

center  of,  122 

flow  of  liquids,  196 

pendulum,  the  simple,  86 

separation  of  cream,  76 
Gridiron  pendulum,  234 
Guillaume,  230 
"Guinea  and  feather"  experiment, 

35 
Gyroscope,  80,  81 

Hardness,  scale  of,  144 
Harmonic  motion,  simple,  82 
Heat,  absorption  of,  297 

a  form  of  energy,  217,  243,  244 

of  combustion,  248 

conduction  of,  286 

conductivity,  288 

effects  of,  219 

evolution,  260 

exchanges,  Prevost's  theory,  296 

from  electricity,  219 

of  fusion,  250 

measurement  of,  243 

mechanical  equivalent  of,  244 

nature  of,  217 

properties  of  water,  253 

quantity,   equation   expressing, 
245 

radiation,  general  case  of,  300 
determining  factors  in,  296 
through  glass,  298 

reflection,  transmission,  and  ab- 
sorption, by  glass,  300 


INDEX 


341 


Heat,  sources  of,  218 

specific,  244 

transfer,  three  methods  of,  283 

units,  calorie  and  B.T.U.,  243 

uphill  flow  of,  273,  312 

of  vaporization,  250 

applications,    269,    270,    271 

278,  279,  284 
Heating  system,  hot-air,  283 

hot-water,  284 

steam,  285 

High  altitudes,  boiling  point  at,  261 
"  Hit-or-miss  "  governor,  329 
"Holes"  in  the  air,  aeroplane,  52 
Hooke's  law,  147 

Hoop,  kinetic  energy  of  translation 
and  rotation  are  equal,  98 
Horizontal  beams,  strength  of,  150 
Horse  power,  of  engines,  106,  107 

French,  105 

hour,  106 

value  of,  105 
Hotbed,  the,  299 
Hot-water  heating  system,  284 
Hourglass,  the,  10 
Hurricanes  and  typhoons,  308 
Hydraulic  elevator,  206 

press,  206 

ram,  207 
Hydraulics,  general  discussion,  194, 

195 

Hydrogen  thermometer,  227 
Hydrometers,  167 
Hydrostatic  paradox,  161 

pressure,  158 
Hygrometer,  chemical,  303 

wet-and-dry-bulb,  269,  303 
Hygrometric  tables,  304 
Hygrometry,  303 

Icebergs,  origin  of,  258 

flotation,  165 

Ice  calorimeter,  Bunsen's,  251 
density  of,  140,  165 
-cream  freezer,  258 
lowering   of   melting   point   by 

pressure,  256 

manufacture  of,  ammonia  proc- 
ess, 271 


Ice,  manufacture  of,  can  system,  273 

plate  system,  273 

"Ideal"  engine,  Carnot's,  313,  323 
Ignition  temperature,  220 
Immersed  floating  bodies,  164 
Impact  of  bodies,  52 
Impulse  equal  to  momentum,  52 
Impulsive  force,  52 
Inclined  plane,  117 

mechanical  advantage  of,  118 
Indicator,  319 

card,  gas  engine,  330,  331 
use  of,  321 

diagram  or  "card,"  320 
Induction  coil,  gas  engines,  327 
Inertia  force,  49 

torque,  driving,  69 

work  done  against,  89,  93 
Injector,  steam  boilers,  211,  212 
Interference  of  sound  waves,  292 

of  light,  293 

Intermolecular  attraction,  work 
against,  surface  tension,  172 
Internal  work  done  by  gas  in  expand- 
ing, 278 

Interpolation,  49 
Invar,  230 

Inverse  square,  law  of,  31 
Isothermal  compression  and  expan- 
sion, 324,  325 

lines,  Carnot's  cycle,  324 
Isothermals  of  a  gas,  188,  190 

of  carbon  dioxide,  274,  276 

Jackscrew,  the,  120 
Jet  and  ball,  212 

condenser,  316 

pump,  209 

Joly's  steam  calorimeter,  252 
Joule,  James  P.,  277 

unit  of  energy,  90 
Joule's  determination  of  mechanical 

equivalent  of  heat,  218 
Joule-Thomson  experiment,  277 

Keokuk,  water  power,  205 
Kilogram,  4 
Kilowatt-hour,  106 
Kilowatt,  the,  106 


342 


INDEX 


Kindling    or    ignition    temperature, 

220 
Kinetic  energy,  92,  96 

and  perpetual  motion,  93 

units  of,  95 
Kinetic  theory  of  evaporation,  260, 

261 
of  gases,  236 

and  Boyle's  law,  188 
of  gas  pressure,  179,  188 
of  heat,  217 
of  matter,  138 
Lamp,  Davy's  safety,  287 

the  "skidoo,"  232 
Land  and  sea  breezes,  306 
Law,  of  Boyle,  187,  192 
of  Charles,  236 
of  cooling,  Newton's,  297 

Stefan's,  297 
Dulong  and  Petit's,  246 
of  gases,  general,  240 
of  gravitation,  Newton's,  30 
of  inverse  square  of  distances,  31 
of  Pascal,  205 

Laws,  of  falling  bodies,  38-48 
of  friction,  100 
of  gases,  three,  239 
Newton's  three,  of  motion,  49 
Length,  measurement  of,  5 
standard  of,  2,  4 
unit  of,  2,  4 
Lever,  "arm,"  60 

"resistance    arm"    and    "force 

arm,"  114 
three  classes  of,  113 
the  compound,  130 
Light,  visible,  ultra-violet,  and  infra- 
red, 291 

interference  of,  293 
Linde's  liquid  air  machine,  280 
Linear  expansion,  228 
coefficient  of,  229 
relation  to  coefficients  of  cubical 
expansion     and     area    ex- 
pansion, 235 
"Line  of  centers,"  126 
Liquefaction  of  gases,  278-282 

"cascade"  or  series  method,  279 
regenerative  method,  280 


Liquid  air,  279,  280 

properties  and  effects  of,  281 
Liquids,  density  of,  140,  161 

elasticity  of,  155 

high  velocity — low  pressure,  208 

properties  of,  155 

specific  gravity  of,  167 

transmission  of  pressure  by,  159 

velocity  of  efflux,  196 
Locomotive,  maximum  pull  of,  102 
"Loss  of   weight"  in  water,  Archi- 
medes' principle,  163 
Low  "area,"  in  cyclones,  306 

Machine,  defined,  110 

efficiency  of,  111,  112 

liquid  air,  279,  280 

perpetual  motion,  93 

simple,  112 

theoretical  and  actual  mechan- 
ical advantage  of,  111 
Malleability,  144 
Manometer,  closed-tube,  191 

open-tube,  191 

vacuum,  193 

Marriotte's  or  Boyle's  law,  187 
Mass,  center  of,  124 

definition  of,  8 

and  inertia,  8 

measurement  of,  8 

and  weight  compared,  8 
Matter,  conservation  of,  139 

divisibility  of,  138 

general  properties  of,  139 

kinetic  theory  of,  138 

structure  of,  138 

three  states  of,  137 
Maximum  density  of  water,  255 

and     minimum     thermometer, 
Six's,  226 

thermometer,  225 

Mean  free  path,  of  gas  molecules, 
139 

solar  day,  3 

Measuring  microscope,  or  microm- 
eter microscope,  7 

Mechanical  advantage,   actual  and 
theoretical,  111 

equivalent  of  heat,  244 


INDEX 


343 


Melting  point,  255 

of  alloys,  255 

effect  of  pressure  on,  256 

table  of,  256 
Meniscus,  223,  277 
Mercury,  air  pump,  201 

barometer,  the,  184 

boiling  point,  222 

freezing  point,  222 

merits    for    therm  ometric    use, 

222 
Mercury-in-glass  thermometer.  222 

calibration  of,  223 

filling  of,  222 

fixed  points  on,  223 
Metal  thermometer,  227 
Meteorology,  302 
Meteors,  cause  of  glowing,  181 

and  height  of  atmosphere,  181, 
Method   of  mixtures,   specific  heat 

determination  by,  247 
Metric  system,  the,  4 
Micrometer  caliper,  6 

microscope,  7 
Moduli,  the  three,  152 
Modulus,  of  shearing  or  rigidity,  152 

of  tension,  Youngs,  147 

of  volume  or  bulk,  152 
Moisture  in  the  atmosphere,  302 
Molecular   freedom,    solids,    liquids 
and  gases,  138 

motion,  kinetic  theory  of  gases, 

236 

in  heat,  vibratory,  217 
Molecules  and  atoms,  138 

of  compound,  138 

"surface"  and  "inner,"  169 
Moment  of  inertia,  defined,  66 

of  disc  and  sphere,  68 

of  flywheel,  approximate,  68 
practical  applications  of,  68 

value  and  unit  of,  67 
Momentum,  conservation  of,  52,  53, 
54,  55 

defined,  52 

equals  impulse,  52 
Monorail  car,  82 

Moon,   gravitational    attraction   on 
the  earth,  33 


Moon,  path  of,  32 

production  of  tides  by,  33 
Motion,  accelerated,  28 

of  falling  bodies,  38 

heat,  a  form  of,  217 

Newton's  laws  of,  49 

non-uniformly  accelerated,  29 

perpetual  versus  the  conserva- 
tion of  energy,  93 

planetary,  32 

of  projectiles,  42,  43,  44 

rotary,  59-71 

screw,  24 

of  a  ship  in  a  rough  sea,  24 

simple  harmonic,  82 

translatory,  23-58 

uniform,  28,  29 
circular,  72 

uniformly  accelerated,  28,  29 

wave,  290 

Nature  of  heat,  217 
Negative  acceleration,  25 

torque,  60 
Neutral  equilibrium,  126 

layers,  strength  of  beams,  150 
Newton's  gravitational  constant,  31 

law  of  cooling,  297 
of  gravitation,  30 

laws  of  motion,  49 
Nickel-steel  alloy,  invar,  230 
Nimbus,  or  rain  cloud,  302 
Numeric  and  unit,  2 

Olzewski,  liquefaction  of  gases,  279 
Onnes,  low  temperature  work  of,  237 
Orchards,    "smudging    of"    during 

frost,  299 
Osmosis,  157,  158 
Osmotic  pressure,  157 
"Outer  fiber, "  strength  of  beams,  150 

Pascal,  French  physicist,  185 
Pascal's  law,  205 
Pendulum,  ballistic,  55 

compensated,  234 

gridiron,  234 

simple  gravity,  86 

torsion,  the,  87 


344 


INDEX 


Period  of  pendulum,  86 

in  simple  harmonic  motion,  85 
Permanent  set,  elasticity,  149 
Perpetual  motion,  93 
Physical  quantity,  definition  of,  1 
Physiological    effect    of    heat,    219, 

222 

Pictet,  liquefaction  of  gases,  278 
Pitch,  in  music,  293 

of  a  screw,  7 
Planetary  motion,  32 

direction  of  rotation,  34 
Plastic  substances,  142 
Platform  scale,  130,  131 
Platinum,  why  used  in  sealing  into 

glass,  230 

Plotting  of  curves,  48 
Polygon  of  forces,  12,  16 

vector,  closed,  15 

Porous  plug  experiment,  the  Joule- 
Thomson,  277 
Potential  energy,  92,  95 
Pound  mass,  and  pound  weight,  2, 

27,36 

Poundal,  27,  36 
Power,  denned,  104 

of  engines  and  motors,  by  brake 

test,  107 

in  linear  motion,  104 
in  rotary  motion,  106,  107 
of  steam  engine,  317 
transmitted  by  a  shaft,  154 
units  of,  105,  106 
Precession  of  equinoxes,  82 

in  gyroscope,  81 

Precipitation,  rain,  snow,  etc.,  302 
Pressure,  atmospheric,  183,  199 

diminution  of  in  regions  of  high 

velocity,  208 
effect  on  boiling  point,  262 

on  freezing  point,  256 
exerted  by  a  gas,  kinetic  theory, 

179 

gage,  Bourdon,  192 
gradient,  and  temperature  gra- 
dient compared,  289 
perpendicular    to    walls,     161, 

184 
steam  gage,  192 


Pressure  of  saturated  vapor,  262 
aqueous  vapor,  table,  263 

transmission  by  liquids,  159 
Prevost's  theory  of  heat  exchanges, 

296 
Principle  of  Archimedes,  163 

of  d'Alembert,  49,  51 
Projectiles,     drift     due    to    earth's 
rotation,  307 

maximum  height  reached,  46 

motion  of,  42,  43,  44 

range,  and  maximum  range,  47 

velocity  and  air  friction,  46 
Projection,  meaning  of,  83 
Prony  brake,  106,  107 
Properties  of  fluids  in  motion,  194- 
214 

of  gases  at  rest,  177-193 

of  liquid  air,  281 

of  liquids  at  rest,  155-176 

of  matter,  general,  139 

of  saturated  vapor,   266,   267, 
268 

of  solids,  144-154 
Pulley,  the,  114 
Pulleys,    "fixed"    and    "movable," 

115 
Pump,  air,  200 

centrifugal,  204 

force,  200 

Geryk,  201 

jet,  209 

rotary,  203 

Sprengel,  201 

suction,  198 

turbine,  204 

Quantity  of  heat,  measurement  of, 

unit  of,  243 
physical,  defined,  1 

Radial  acceleration,  73 
Radian,  the,  62 
Radiant  heat,  296 
Radiation,  295 

and  absorption,  297 
Rainfall,  where  excessive,  305 
Rain,  snow  and  other  precipitation, 
302 


INDEX 


345 


Range  of  projectiles,  45 
Reaction,  of  aeroplane,  52 
of  birds,  wings,  52 
practical    applications    of,    51, 

202 

of  propeller,  51 
in  swimming,  51 
in  traction,  51 

Reaumer  thermometric  scale,  225 
Receiver,  the,  179 
Recording  thermometer,  227 
Reflection  and  refraction  of  waves, 

293 
Refraction,  294 

makes  vision  possible,  295 
produces  rainbow  and  prismatic 

colors,  295 
Refrigerating  apparatus,  ammonia, 

271,  272 

Refrigerator  room,  273 
Regelation,  257 
Regenerative  method,  of  liquefying 

gases,  280 

"Resistance  arm,"  of  levers,  114 
Resisting  force,  F0,  simple  machines, 

110 
Resolution,  forces  into  components, 

19,  101 
of  vectors,  19 

Restitution,  force  of  in  simple  har- 
monic motion,  83,  84,  85,  86 
Resultant  of  several  forces,  11,  12, 

13,  16 
defined,  11 
torque,  61 
Rifle  ball,  velocity  at  various  ranges, 

46 
velocity  by  ballistic  pendulum 

method,  55 
flight  of,  44,  45,  46 
Rigid     body,     two     conditions     of 

equilibrium  of,  64 
Rigidity,  modulus  of,  152 

of  shafts 

Rocking  chair,  equilibrium  with,  126 
Rolling  friction,  102,  103 
Rose's  metal,  255 
Rotary  blowers  and  pumps,  203 
motion,  59 


Rotary  motion,  uniformly  acceler- 
ated,    and    non-uniformly 
accelerated,  59 
and  translatory  motion,    for- 
mulae compared,  70 

Rotor    and    stator    vanes,     steam 
turbine,  322 

Rumford,      Count,      cannon-boring 
experiment,  217 

Safety  lamp,  Davy's,  287 
Sailing  against  the  wind,  20 
faster  than  the  wind,  21 
Saturated  solution,  156 
vapor,  261 

pressure,  262,  268 
properties  of,  266 

table  of,  263 
Scalars  and  vectors,  11 

addition  of,  compared,  12 
Scale,  platform,  130,  131 
Screw,  the,  120 

propeller,  204 
Sea  breeze,  306 
Second,  defined,  2 
Sensitiveness  of  beam  balance,  128 
defined,  7 

of  micrometer  caliper,  7 
of  vernier  caliper,  6 
Shafts,  rigidity  of,  153 

power  they  can  transmit,  153 
Shearing  stress,  strain  and  elasticity, 

151,  152 

Ship,  motion  of  in  a  rough  sea,  24 
Shrinking  on,  or  setting  of  wagon 

tires,  228 
Sidereal  day,  3 
Simple  harmonic  motion  (S.  H.  M.)i 

82 

Simple  machines,  the,  112 
efficiency  of,  111,  112 
inclined  plane  type  and  lever 

type,  121 

mechanical  advantage  of,  111 
Simple  gravity  pendulum,  86 
Siphon,  the,  197 

Six's  maximum  and  minimum  ther- 
mometer, 226 
Skate,  "bite"  of,  257 


346 


INDEX 


"Skidoo"lamp,  232 

Slide  valve,  steam  engine,    318 

Slug,  the,  37 

"Smudging"  of  orchards,  protection 

against  frosts,  299 
Snow,  rain  and  other  precipitation, 

302 
Soap   films   tend   to  contract,   171, 

172 
Solar  day,  mean,  3 

variation  of,  3 
heat,  power  of  per  square  foot, 

218 

motor,  S.  Pasadena,  Cal.,  296 
Solids,  thermal  conductivity  of,  288 
density  of,  140 
elasticity  of,  145-151 
properties  of,  145-154 
Solution,  boiling  point  of,  255 
freezing  point  of,  262 
of  solids,  liquids  and  gases,  156 
of  metals,  amalgams,  159 
saturated,  156 
Sound  waves,  290 

interference  of,  292 
Sources  of  heat,  218 
"Spark"  coil,  328 

Specific   gravity,   by  balanced   col- 
umns, 162 
defined,  166 
hydrometer  scale,  168 
of  liquids,  167 
of  solids,  167 
Specific  heat,  defined,  244 

method  of  mixtures,  247 

table  of,  245 

the  two  of  gases,  246 

the  ratio  of  the  two  of  gases, 
246 

of  water,  243 
Speed,   average,    24 

and  velocity  compared,  24 
Sphere  of  molecular  attraction,  169 
Spinney,  L.  B.     See  Preface. 
Sprengel  air  pump,  201 
Spring  balance,  130 

gun  experiment,  47 
Stable,  unstable  and  neutral  equi- 
librium, 126 


Standards  of  length,  mass  and  time, 

2,4 

kilogram,  4 
meter,  4 
pound,  2 
yard,  2 
Steam  boiler,  efficiency  of,  312 

calorimeter,  Joly's,  252 
Steam  engine,  311,  314,  319 
compound,  315 
condensing,  316 
efficiency  of,  312,  313 
governor,  315 
indicator  card,  319,  321 
methods    of    increasing    effi- 
ciency of,  315 
power  of,  317 
thermodynamic  efficiency  of, 

313 

triple  expansion,  315 
work  per  stroke,  316 
Steam  pressures  and  temperatures, 

table  of,  274 
Steam  turbine,  205,  321 

advantages  of,  321 
Steel,    composition    of    and    elastic 

properties  of,  149 
Steelyard,  the,  129 
Stefan's  law  of  cooling,  297 
Stiffness  of  beams,  150 
Strain,  three  kinds  of,  151 

tensile,  146 
Strap  brake,  108 
Stress,  tensile,  147 

three  kinds,  151 

Stretch  modulus,  or  Young's  modu- 
lus, 147 

Sublimation,  260 
Suction  pump,  198 
Supercooling,  256 
Superheating,  256,  265,  266 

of  steam,  315 

Surface  condenser,  steam,  316 
Surface  a  minimum,  surf  ace  tension, 

170,  171 
Surface  tension,  and  capillarity,  168- 

175 

defined,  171 
effects  of  impurities  on,  173 


INDEX 


347 


Surface  tension,  methods  of  measur- 
ing, 172,173,  175 
value  for  water,  172 
Systems  of  measurement,  British,  2 
metric,    4 


Table  of  boiling  points,  262 

coefficient  of  linear  expansion, 

230 

of  cubical  expansion,  235 
of    critical    temperatures    and 

critical  pressures,  274 
of  densities,  140 
of  heats  of  combustion,  249 
of  heats  of  fusion,  251 
of  heats  of  vaporization,  251 
hygrometric,  304 
of  melting  points,  256 
of  saturated  vapor  pressure  of 

water,  263 
of  specific  heat,  245 
of  thermal  conductivity,  289 
Temperature,  absolute,  236 

compensation,  watch  and  clock, 

233,  234 
critical,  273 
defined,  220 
gradient,  289 
sense,  221 
of  the  sun,  298 
scales,  absolute,  237 
centigrade,  224 
Fahrenheit,  224 
Reaumer,  225 
sense,  221 

Tensile  strength,  144,  148 
Theorem  of  Bernoulli,  209 

of  Torricelli,  196 
Theoretical  mechanical   advantage, 

111 

Thermal  capacity,  244 
conductivity,  288 
conductivities,  table  of,  289 
bottle,  Dewar  flask,  282 
Thermobattery,  242 
Thermocouple,  the,  241 
Thermodynamic    or    limiting    effi- 
ciency, engines,  313 


Thermodynamics,  311 

first  law,  statement  of,  311 

illustration  of  first  law,  311 

second  law,  statement  of,  312 
Thermograph,  227 
Thermometer,  calibration  of,  223 

centigrade,  224 

clinical,  225 

dial,  227 

gas,  constant  pressure,  226 
constant  volume,  226 

hydrogen,   constant  volume,   a 
standard,  227 

maximum,     of     Negretti     and 

Zambra,  225 
and  minimum,  Six's,  226 

metallic,  227 

mercury-in-glass,  222 

recording,  227 

wet-and-dry-bulb,  269 
Thermometry  and  expansion,  217 
Thermopile,  242 
Thermostat,  231,  300 
Thomson,  Sir  Wm.   (Lord  Kelvin), 
plug  experiment,  277 

statement    of    second    law     of 

thermodynamics,  312 
Three  states  of  matter,  137 
Tides,  cause,  spring  and  neap,  34 

lagging  of,  34 

in  Bay  of  Fundy,  34 
Time,  of  flight  and  range,  45 

measurement  of,  9 

measurer,  essentials  of,  9 

spacing  and  spacers,  9,  10 

standard  of,  mean  solar  day,  3 

unit  of,  2,  4 
Tornadoes,  309 

extent,  310 

origin,  309 

pressure  in,  310 

velocity  of,  310 
Torque,  59,  60,  61 

accelerating,  66 

driving  inertia,  69 

positive  and  negative,  60,  61 

resultant,  61 
Torricelli's  experiment,  185 

theorem,  196 


348 


INDEX 


Torsion  pendulum,  87 
Trade  winds,  305 

Transfer  of  heat,  three  methods,  283 
Transformation  of  energy,  93 
Transmission    of   heat  radiation 
through  glass,  298 

of  pressure,  159 
Transverse  wave,  292 
Triple  expansion  engine,  315 
"Tug  of  war,"  forces  in,  50 
Turbine  pump,  204 

water  wheel,  205 
Twilight,  cause  of,  181,  182 
Typhoons,  308 

Uniform  circular  motion,  72 
central  force  of,  72 
centrifugal  force  of,  72,  74 
radial  acceleration  of,  73,  74 
Uniform  motion,  linear,  28 

rotary,  59 
Units,  absolute,  or  C.  G.  S.  system,  4 

of  acceleration,  26 

British  system,  2 

conversion  of,  4 

of  force,  27,  36 
and  weight,  36 

fundamental,  2 

of  heat,  243 

of  mass,  2,  4 

of  moment  of  inertia,  67 

and  numerics,  2 

of  power,  105 

of  time,  2,  4 

of  work,  90 
Universal  gravitation,  30 

Vacuum,  185 

cleaner,  203 

gage,  193 

pans,  264 
Vapor  and  gas,  distinction  between, 

277 

Vapor  pressure  of  water  at  different 
temperatures,  table,  263 

saturated,  261 
Vaporization,  cooling  effect  of,  268 

denned,  260 

heat  of,  250 


Vaporization  table,  251 

two  opposing  tendencies  in,  266 
theory  of,  261 
Vector  addition,  12 
defined,  11 
equilibrium,  15,  18 
graphical  representation  of,  12 
polygon,  closed,  represents  equi- 
librium, 15 
resolution  of  into  components, 

19 

scale  for,  12 
triangle,  closed,  15 
Velocity,  acquired,  38,  39 
angular,  62 

and  linear  compared,  63,  70 
dependent  upon  vertical  height 

of  descent  only,  55 
average,  24,  38,  39,  40 
of  efflux,  196 
of  falling  bodies,  38 
head,  194,  196 

initial,  final  and  average,  38,  39 
of  rifle  ball,  at  different  ranges, 

46 

by  ballistic  pendulum,  55 
versus  speed,  11,  24,  25 
Velocities,  addition  of,  13 
polygon  of,  15 
relation  of  in  impact,  53 
resolution  of  into  components,  19 
resultant  of,  13,  14 
"steam,"  "drift,"    and    "walk- 
ing," 14,  15. 

Venturi  water  meter,  211 
Vernier  caliper,  5 

principle,  6 
Vibration,    direction    of    in    wave 

motion,  292 
in  simple  harmonic  motion,  82, 

83 

Viscosity  of  liquids,  155 
of  gases,  177 

and  the  kinetic  theory,  177 
Volume,  change  of  with  change  of 

state,  256 
elasticity  of,  151 
modulus,  152 
strain,  152 


INDEX 


349 


Wagon,  hillside,  127 

Water,  compressibility  of,  155,  165 

critical  temperature  of,  273 

density  of  in  British  system,  140 

freezing    point    variation  with 
pressure,  256 

maximum  density  of,  255 

meter,  Venturi,  211 

peculiar  thermal  properties  of, 
253,  254 

waves,  290 

reflection  of,  294 
Watson,  W.     See  Preface. 
Watt,  unit  of  power,  106 
Watt-hour-meter,  106 
Watt's  centrifugal  governor,  79,  315 

indicator  card  or  indicator  dia- 
gram, 320 

Wave  length  of  ether  waves,  291, 
292 

motion,  290 

direction  of  vibration  in,  292 
longitudinal    and    transverse 
vibrations  in,  292 

trains,  interference  of,  292 
Waves,  actinic,  291 

ether,  291 

heat,  291 

Hertz,  291 

light,  291 

reflection,  293 

refraction,  294 

sound,  290 

water,  290 
Weather  bureau,  service  of,  187 

predictions,  187 
Wedge,  the,  118 

and  sledge,  119 
Weighing  machines,  30, 127 

the  earth,  30 


Weighing,  process  of,  127 
Weight  compared  with  mass,  8 

in  a  mine,  30 

variation  of  with  altitude  and 
latitude,  9,  35 

units  of,  36 
Welding,  141 

Wet-and-dry  bulb  hydrometer,  296 
Wheel  and  axle,  117 
Windlass,  111 
Windmill,  reaction  in,  202 
Winds,  304 
Wood's  metal,  255 
Work,  defined,  89 

done  by  a  torque,  92 

of  driving  inertia  force,  90 

in  forming  liquid  film,  172 

against  friction  produces  heat, 
99,  311 

involved  in  all  energy  transfor- 
mations, 93,  94 

if  motion  is  not  in  the  direc- 
tion of  force,  91 

obtained  from  heat,  311 

from    water  under  pressure, 
210 

per  stroke  of  steam  engine,  316 

units  of,  90 

used  in  three  ways,  89,  90,  93 
Working  force,  110 


Yard,  standard,  2 
Yield-point,  148,  150 
Young's  modulus,  147 

Zero,  absolute,  236 

change  of,  with  age  of  thermom- 
eter, 224 
Zone  of  calms,  305 


I' 


BH^ 


Wfcv        «$t 


0-jtf 


nfe     £«mi 


:lOS-ANCn&, 


Jftv 


ITVD-JO^ 


IIVD-JO^ 


slOS-ANCEl^        ^OF-CAUFO^     ^OF-CAIIFOR^ 

S 


^Aavaan^     youmtfi 


a% 

ci 
nl 

in-mvN 

3 
^1 

w-a^ 

Mflftfe 


an-^ 
m*£ 

cj 

^i§ 

fflfl^ 

ICflft^ 


B 


•Ill 

L  005  489  477  9 


<lOS-ANGEl£f£ 


£§2™!LREglONAL  LIBRARY  FACILITY 


fl 
coe 


AA    000710680    o 


